What is the explanation for the interference patterns in MWI? In Young's double-slit experiment, MWI states that in some "worlds" the particle goes through one slit, and in others it goes through the other. If this is so, why do we get an interference pattern? The particle is not interacting with itself and is defined by classical mechanics?
Furthermore, is MWI deterministic? If it is, how is it even possible for there to be many worlds? Surely the state of the world at a given time + the laws of physics will always result in the same future space-time in any deterministic theory? 
 A: Firstly, I find the hostility against many worlds interpretation inadequate. As far as understanding quantum mechanics goes, the case is far from closed. I believe that it is unlikely, that this question will be answered near future (and it is plausible that it will be never resolved). Nevertheless, something being inherently hard should not suppress our thinking.
Secondly, why do I believe that many worlds interpretation is a sound approach? Before even hearing the MWI-word, we were talking about quantum entanglement in our department coffee room's blackboard (some 7 years ago) and I realized that measurement can be explained by quantum entanglement of the observer and the system. I think most physicists are too busy thinking about real problems (as they should), that they never sit down to think about these fundamental issues. (Also, one of the reasons is probably the choice of name and the stupid splitting movie film picture in Wikipedia).
Thirdly, I will have do define carefully what MWI is. As is correctly stated in the other answers, there are various definitions (the vicious say that there are as many definitions of MWI as there are supporters):
First thing you have to understand about MWI that it is formulated by a PhD student 60 years ago. The decoherence will not be invented in 15 years and the current main-stream interpretation of quantum mechanics is the Copenhagen interpretation with unintuitive wave function collapse postulate. Still, I find the thesis to be quite remarkable (Everett is also cited by decoherence paper, but I cannot access it so I do not know if it is in good or bad).
In Everett's thesis, he defines two ways of changing the wave function (as listed by von Neumann).


*

*Process: Suddenly, by assigning the system into eigenstates with probabilities $|<\phi_i|\Psi>|^2$.

*Process: Via unitary evolution, according to Schrodinger equation.


He lists several alternatives, but sticks with alternative 5:

To assume the universal validity of the quantum description,
  by the complete abandonment of Process 1. The general
  validity of pure wave mechanics, without any statistical assertions,
  is assumed for all physical systems, including observers and measuring
  apparata. Observation processes are to be described completely
  by the state function of the composite system which includes
  the observer and his object-system, and which at all times
  obeys the wave equation (Process 2).

As far as I see it, this is the crux of many-worlds interpretation. There is no need to consider a) unexplained sudden changes to wave function b) probabilistic rules of resetting quantum simulations at certain moments. Here a) and b) differ by whether the collapse is real in other interpretations.
Here is another quote:

We have seen that in almost all of these observer states it appears
  to the observer that the probabilistic aspects of the usual form of quantum
  theory are valid. We have thus seen how pure wave mechanics, without
  any initial probability assertions, can lead to these notions on a subjective
  level, as appearances to observers.

Here is one more:

We have shown that our theory based on pure wave mechanics, which
  takes as the basic description of physical systems the state function -
  supposed to be an objective description (i.e., in one-one, rather than
  statistical, correspondence to the behavior of the system) - can be put in
  satisfactory correspondence with experience. We saw that the probabilistic
  assertions of the usual interpretation of quantum mechanics can be
  deduced from this theory, in a manner analogous to the methods of classical
  statistical mechanics, as subjective appearances to observers -
  observers which were regarded simply as physical systems subject to the
  same type of description and laws as any other systems, and having no
  preferred position. The theory is therefore capable of supplying us with
  a complete conceptual model of the universe, consistent with the assumption
  that it contains more than one observer.

As far as I interpret this, Everett is saying that probabilistic interpretation of quantum mechanics is emergent from properties of unitary evolution of the wave function. This is appealing in many ways. First of all, wave function collapse as a phenomenon is explained. It is an entanglement between observer and the system. When added with decoherence, one has a theory of measurement which is complete and contains no awkward collapse postulates.
Finally and foremost, and this is why all the fuzz, here comes the many worlds part. Pure wave mechanics, also for observers, means that there are finite amplitudes in many different observer states simultaneously. Thus, if with a world we mean all things we can get information about, we will see, that the scientist who measured spin up will never communicate with the scientist who measured spin down. With orthodox interpretation, both of these scientists still exists in the world wave function.
Now, finally, and unfortunately at so late part of this answer because of all the fuzz, we can come to your questions:

MWI states that in some "worlds" the particle goes through one slit, and in others it goes through the other. If this is so, why do we get an interference pattern?

It does not state that at all! Exactly the opposite. It states all the regular things about two-slit experiment which can also be stated with probabilistic interpretations: Only, if one measured from which slit the particle goes, one does not get the interference pattern. Probabilistic interpretations call this collapse. MWI says that there is a quantum entangled state $|a>|A>+|b>|B>$ where small letters are slits and capitals observers reading a or b from their measuring apparatus.

Is MWI deterministic?

Yes, everything will be governed by the unitary evolution of the wave function. The Schöringer equation is of first order in time and thus exactly solvable when given a boundary condition (say wave function at surface t=0). This means that the system is fully deterministic. To elaborate further, in many worlds interpretation the unitary evolution of the wave function produces collapse of wave function only as an emergent process (thus abadoning process 1, as listed above).
To conclude, MWI should be treated as a historical way to modern understanding of quantum mechanics and measurement. In it's 60's formulation it is outdated, but the concepts still hold. I find it funny, that the proponents of probabilistic theory seem to find the MWI-theory non-sense, even their own interpretation can be derived MWI. It remains to be seen if future experiments can shed light into these interpretations. For example, although very unlikely, it would be very exciting if the progress in quantum computing would hit unexplained mysterious limitations requiring new theories. As far as the dislike towards this theory goes, it is probably related to fact that discussing interpretations is mostly hobby to everyone. There are very few who actually does research on this fields, and could comment on the latest events. The rest are probably like me: when I go to work tomorrow, I will wonder about theoretical modelling of plasmonics in photovoltaics, since that is what I get paid to do.
A: As stated by Luboš Motl in another answer, there is no consensus between Everett's interpretation contenders about what it means exactly. The common idea is that no state evolution other than unitary as per Schrödinger should be accepted (no collapse) but that's about it.
Indeed it is not clear at all what is supposed to be splitting or branching, and when. For a thorough review of this topic ("many what, exactly"?), see 
Multiplicity in Everett's interpretation of quantum mechanics (Louis Marchildon, 2015).
A: There doesn't exist any "totally well-defined" realization of MWI but its champions want to agree with the basic experimental facts so they would almost certainly say that there is no splitting of the worlds when a particle goes through two slits.
Instead, if there's any splitting of the worlds at all, and different MWI advocates have different opinions whether it occurs at all (in particular, Everett's opinion was No – and he wrote an expletive on a document by DeWitt who introduced "many worlds" for the first time) – the splitting only occurs at the moment when many degrees of freedom decohere or a human observation is made, e.g. after the particle hits the photographic plate at one point or another. (Again, there is no well-defined description of the moment when the splitting is supposed to take place, or how it takes place etc.)
Before that, MWI assumes that the wave function "objectively exists" and its parts of the wave function going through the two slits interfere with each other just like they interfere in standard quantum mechanics.
MWI is meant to be a deterministic theory where the wave function is an object that objectively exists, a set of classical degrees of freedom, and that evolves according to a deterministic Schrödinger equation. There is no contradiction between having many worlds and determinism. However, there may be a contradiction between assumptions of MWI and other observed facts.
Most contemporary MWI advocates imagine that the "many worlds" are nothing else than the "parts" of the wave function that are widely separated in the configuration space (a power of the real space). No universal set of rules how the wave function may be divided to parts exists but when the evolution of the parts get "macroscopically different", it starts to make sense to say that "two portions of the wave function are hugely separated".
A: In the many world interpretation of quantum mechanics, there is only a split into different branches when decoherence takes place. So there would be no split in the two slit experiment until the photon hits the photographic plate. At that stage decoherence takes place and there would be a split into a separate Universe (branch) each of which has the photon hit a different part of the photographic plate. The theory is deterministic  until you try to answer what is the probability of being in the different Universes. This is then taken as being proportional to the amplitude squared of the value of the wave function for each Universe.
There is no widely accepted  way of experimentally distinguishing the many worlds interpretation from the more traditional Copenhagen interpretation. So the question which is more correct is more of a philosophical one rather than a physics one.
A: The Many Worlds Interpretation is a theory of non relativistic quantum mechanics where there is a wavefunction from the configuration space of the entire system (and this is utterly essential) into the joint spin state of the entire system and it evolves according to the Schrödinger equation, and nothing else.
No one claims (or has ever claimed) that in one world the particle goes through one slit and in another world the particle goes through the other. Instead, what happens is the wavefunction of the system evolves according to the Schrödinger equation, hence, according to the Hamiltonian.
If you want to see why interference happens, it helps to first contrast with a situation where there is no interference, e.g. one where you get which-way information.
We will have the same setup for both situations. So for instance, the x axis could represent the x position of particle one and the y axis could represent the $y$ position of particle one and the z axis could represent the position of particle two. Then the wavefunction must assign a value to each combination of the locations of each particle (the configuration space is $\mathbb R^{3n}$ and a single point $ (\vec r_1, \vec r_2, ... , \vec r_n)$ tells you a classical configuration of every particle).
But your wavefunction assigns values to every possible configuration. Possibly zero. Possibly nonzero. Let's have the wavefunction have its complex phase oscillate in the x direction, but be confined to a finite spread in the x direction (confined near $x=-1$). The oscillation of phase in the x direction means the region of support (where it has nonzero values) will move in the positive x direction. Let's also focus it in the z direction so it has a finite spread in the z direction (confined near $z=0$). But in the y direction it will be bimodal. It will have a region near $y=-10$ where it is nonzero, and a region near $y=+10$ where it is nonzero. But go a little bit from those values and it drops to zero.
So it's like you had a packet moving in the x direction and focused in the x direction and focused in the z direction and also focused in the y direction near $y=-10$ and then you had a second packet moving in the x direction and focused in the x direction and focused in the z direction and also focused in the y direction near $y=+10$. Your initial wave is the sum of those packets, so it is nonzero in both those regions of configuration space.
But those regions aren't worlds.
Now, if you go through a slit with a which way detector, then the wave confined near $y=-10$ goes through a slit with center at $x=0$ and $y=-10$ but because of the which way detector, the wave is deflected in the direction $-\hat z$ so even as it spreads out on the $y$ direction it is systematically deflected in the $-\hat z$ direction.
So it eventually hits a screen at $x=200$ all concentrated at $z=-200$.  It's like if you put a fiber optic cable on the slit and aimed the beam down.
And the wave confined near $y=+10$ goes through a slit with center at $x=0$ and $y=+10$ but because of the which way detector, the wave is deflected in the direction $+\hat z$ so even as it spreads out on the $y$ direction it is systematically deflected in the $\hat z$ direction.
So it eventually hits a screen at $x=200$ all concentrated at $z=+200$.  It's like if you put a fiber optic cable on the slit and aimed the beam up.
If you actually put fiber optic cables in your slits and sent classical light through it, you'd get a beam deflected down from one slit and a beam deflected up from the other slit, and you would get single slit dictation pattern from each slit.
Both classically and quantum mechanically.
Now let's say there is no which way pattern. Then you wave simply spreads, but isn't deflected. Which means the wave near $y=-10$ spreads out and the wave near $y=+10$  spreads out and by the time they get to $x=+200$ the support of each wave overlaps the support of the other wave.
So it really was one wave with two disjoint regions of support, and the two regions evolved to overlap. When that happens the values interfere and the wave develops parts that have larger values than others.
So far, this is just what the Schrödinger equations says. No interpretations have come into play at all.
The wavefunction gets regions with different sized values solely based on whether those disjoint regions of support evolve to overlap. And they do start to overlap when the location of the particle moving towards the screen isn't causing (by the Hamiltonian) any other particles to move differently.
But when you hit the screen, other particles do start to move differently. The particles at that screen location change. And the particles at the other screen locations do not change.
So when you had the which way detector, you effectively had a screen right there and the waves immediately start to veer away from each other in configuration space. Whereas if you don't have the which way detector they spread and start to overlap before they hit there screen (where they then start to veer away from each other).
Your claim that the particle follows classical mechanics is plain wrong. And MWI doesn't claim that.
In MWI you do have worlds. But many worlds are defined as separate wavepacket whose supports will never overlap with each other in the future. This basically requires that they separately veer in different directions. And since there are $3n$ directions in $\mathbb R^{3n}$ it is easy to veer in different directions once the wave has made many twists in many different particles' directions. And just like two people randomly moving in $3n$ directions. When $n>>10^{24}$ the odds are really bad that they will ever bump into each other. 
So the cutoff of being separate worlds isn't sharp, any more than the size cutoff to use thermodynamics isn't sharp. But eventually when enough particles have been involved, the different twists have placed the support into such different regions that they are not going to overlap again.
So they were not separate worlds when they went through the slit, since separate worlds are defined by not having their support overlap ever again.
The MWI is completely deterministic. There are many worlds because it waves can have regions of support (places where they currently are nonzero) that diverge away from each other in a huge dimensioned space and never overlap again and thus act independently.
There isn't a "classical state of the world." The classical states of the world are literally the points in the huge dimensioned configuration space. And the wavefunction is assigning nonzero vaules to whole regions of configuration space. A world in MWI is a current assignment of nonzero values to a region that evolves in the future as if that is the only place where the values are currently nonzero. You can have multiple worlds. And by definition each acts as if it is the only one.
If you know the definitions it isn't mysterious at all. Start with configuration space. Then assign values to each configuration. Then note that the region where the values are nonzero (the support) sometimes splits into disjoint regions that evolve over time to never again overlap.
Then note that the values in those disjoint regions can act like they are the whole wavefunction and can't tell if they are. Hence it makes sense to call them a world and let each one model itself as the whole world.
If you didn't allow that, you'd be insisting they have to continue modeling parts of the configuration space that don't affect them. For no scientific reason whatsoever. And it wouldn't be wrong per se. It's just extra bookkeeping that doesn't affect that world's predictions. Insisting on modeling things that don't affect your predictions is the domain of people with strong opinions.
People that merely care about making predictions accept that there is a point where it is safe (prediction wise) to simplify things down to have a given world select itself as the only one. Since it won't matter to its predictions about its own future evolution.
