Why are photons particles and waves, instead of particles that simply move in a wave pattern? I understand the double slit experiment, but even with the results, why are photons determined to be particles and waves at the same time, instead of particles that move in a wave pattern? 
In bodies of water observed on Earth, a disruption causes water molecules to move in a direction and those interacting molecules move in a wave pattern. In a perfect system, it seems to me that all water molecules would continue in this motion until the pattern is absorbed. But, water on Earth is not a perfect system, largely due to the effects of gravity. So, are photons described as being particles and waves instead of particles that move in wave patterns constructed from a concept of fluid-dynamics due to a uniform consistency caused by other factors. If this is not totally off the mark, what role would gravity play in this medium? 
 A: Photons are considered by most physicists to be neither waves nor particles, but to exhibit behaviours of both.  In almost every experiment, they exhibit the behaviour associated with one or the other (Shahriar Afshar's 2004 experiment, which purported to show both simultaneously, has had its interpretation contested).
When light undergoes an interaction in which it exchanges energy* (such as hitting a detector), at the moment of interaction it is determined whether it exhibits particle- or wave-like behaviour.
So while light propogates in a wave-like pattern through the electromagnetic field, and is absorbed by electrons as though it is a particle (e.g. the photoelectric effect), it is arguably incorrect to think of light as "being" a wave or a particle, or both.  Its behaviour is only determined after it has interacted and stopped existing.
* You asked about the role of gravity.  When I specified "an interaction in which it exchanges energy", I meant to distinguish from when light appears to bend in a gravitational field, where it does not exchange energy.  Light moves in straight lines; gravity alters the shape of space-time, meaning it is not Cartesian, and straight lines do not behave in the simple way we are used to in a gravitational field.
A: The mainstream approach to quantum mechanics makes particle-like and wave-like characteristics just different manifestations of the same thing. Your approach is similar to Bohmian Mechanics, where the particle definetely has a position but only the "wave associated with it" is accessible to us. The biggest "no-no" of this theory is that it's non-local (information can propagate faster than light, which may violate causality - see Bell's theorem for instance).
Now about photons. Even in the mainstream approach it's kinda hard to say "I measured a photon at position $\mathbf x$"  because when you measure something you inevitably  interact with it, and interacting with photons ends up annihilating them (or creating more photons). The best way to think of a photon is to consider it as a quanta of vibration of the electromagnetic field. The electromagnetic field propagates around and whenever it interacts with, say, electrons it may transfer discrete packets of energy (photons!) to the electron. This is the standard approach of quantum field theory, where we think of all particles as excitations of a corresponding field.
For the last question, there's no medium where the waves propagate. In our modern understanding everything moves in straight lines, but gravity bends space and time and makes those "straight lines" look bent too. Unfortunately (or not?) there are some problems combining general relativity with QM, but we can still measure (classical) gravitational effects on quantum objects. There are many "bouncing ball" experiments where cold neutrons are made to bounce like classical balls under influence of the potential energy
$$
V(z) = mgz.
$$
The agreement of experiment and theory is rather excellent.
A: As you are interested in physics enough to ask this question, I feel I should write a different kind of answer to the above one by cosmogoblin. (I didn't downvote it,  although I think it contains a few points I would argue with, sorry :).

I understand the double slit experiment.

No, you really don't, but that's ok, nobody else does either.......this is because we are trying to understand the quantum world in terms of the classical world.
We build mental pictures of particles as tiny versions of classical particles, such as soccer balls or dust mites, but quantum particles are nothing like anything we can describe in physical terms. 
We also try to explain the property of photons in wave terms, so that we have a chance of predicting (but not understanding) their behaviour when the two slits are open, but if you look closely at the wording of the explanation of the results, you will see that we call the wave a probability wave, and this is definitely not a type of wave that you will see anywhere in the classical world.
My point is that mental pictures are going to let you down when you push them too far in trying to understand the quantum world.
You are, in the long run, far better off trying to get to grips with the mathematical description of a photon and not get caught trying to constantly make sense of quantum behaviour in classical terms. You are not in the classical/ordinary world with these objects.
This is why the idea of the probability wave was developed, as it gives us a way of describing the properties of quantum "things", but it does not try to say what a photon actually is, as we have no idea how to compare it to anything we see around us.

If this is not totally off the mark, what role would gravity play in this medium.

We don't know how to combine gravity with the double slit experiment but, if and when we do, I can pretty much guarantee you that it will totally involve a mathematical description, and no mental picture will be involved.
A: I'll probably get clobbered for this, but I've always found the many-worlds interpretation useful.
The idea is that starting from a particular state in time, all possible world trajectories propagate forward, and the photon particle is a particle and follows a certain path in each world trajectory.
Each world trajectory has a probability amplitude (a complex number), and the trajectories are not independent of each other. They interfere with each other (like waves), and most of them cancel out, meaning their actual probability is near zero, so the photon isn't seen going to weird places.
Where photons are seen is the places where their trajectories have enough probability (due to constructive reinforcement of their amplitudes) to actually happen.
There's more to it, of course...
