What does quantum phenomena exist as prior to observation? It's been said that according to the Schrodinger equation, independent of observation, particles exist in a state of a wave function, which is a series of potentialities rather than actual objects. Then the act of observing, causes a wave of potentiality to collapse to a state of matter.
Is this true? Also, are these quantum experiments anomalies, and not how nature works, or do these quantum experiments, like double slit and quantum erasers, actually reflect how nature works even in the absence of human manipulations.
 A: You speak of the measurement problem, which as yet has no widely accepted resolution.
But a pure quantum state after a measurement is in exactly the same kind of entity as it was before the measurement, namely a new pure quantum state. It's simply that the measurement forces (by whatever as yet less than fully understood mechanism that resolves the measurement problem) the quantum state to be in one of the eigenvectors of the operator that describes the measurement. It is "forced" (collapses ....) into the eigenstate that corresponds to the measurement value gleaned from the measurement. Which eigenstate / vector these are is random - this is the only place where quantum uncertainty enters the picture. From there, it will evolve anew following the Schrödinger equation just as it did before the measurement. One of the outcomes of this last statement is that if the particular measurement operator ("observable") commutes with the Hamiltonian, then the state stays as it is, taking on only a global, linearly-with-time-increasing phase delay. Between measurements, the quantum state evolution is utterly deterministic.
If there is uncertainty in the measurement (i.e. further to and different from the quantum uncertainty above e.g. through inaccuracies of one's instruments), then the state, from the measurer's standpoint, becomes a classical mixture of pure states. In principle, one models this by assigning classical probabilities to each of the pure states that might be consistent with the measurement: these probabilities are the classical conditional probabilities given the particular measurement value that has been gleaned. This is, in fact, a generalization of the situation one has in the Wigner Friend Thought Experiment. In principle one evolves all theses possibilities through parallel Schrödinger equation calculations and then combines their probability predictions classically to calculate overall probability distributions for measurements in the future. In practice, the way to do this calculation more simply is through the density matrix formalism. In short: the underlying dynamics are still those of pure, unitary state evolution: the observer's ignorance about exactly which pure state prevails is encoded into and handled by the density matrix formalism.
A: What you've described is the Copenhagen interpretation, in which the wave function gives you the probability of finding a particle at a certain position. When we observe the particle, we "collapse" the wave function, and then we know where it is.
This isn't the only interpretation, but is the standard one. In general though, it isn't possible to distinguish between different interpretations, because they all give the same experimental results (with a few exceptions).
Let me answer your question "do these experiments actually reflect how nature works?" with another question: if these experiments didn't reflect how nature works, then how could we do these experiments and get these results? In other words, the odd results of these experiments must be consistent with how nature works.
I should point out though that modern interpretations tend to not include the idea of observer and wave function collapse, due to problems like, "what constitutes an observer?" and "why does an observer cause wave function collapse?"
A: 
It's been said that according to the Schrodinger equation, independent of observation, particles exist in a state of a wave function, which is a series of potentialities rather than actual objects. Then the act of observing, causes a wave of potentiality to collapse to a state of matter.
Is this true?

The answer to your first question is: it depends.  :)
The fact that you can take the wavefunction that describes an electron in a potential (one of the basic solutions to schrodinger's equation), and then square this result does indeed give you a probability distribution of said electron in that particular potential.
But you will never know precisely where and with what momentum the electron is without measuring it.  Typically this is done with at least one photon (usually a lot), and the photon is sent towards the probability distribution and the resulting deflected/emitted photon is measured with a detector.  But that measurement is only one very tiny piece of information on where the electron was at the particular moment that the photon interacted with it.  So, in one sense by firing the photon at the electron, you did in fact "fix" the electron state for a brief moment.  However, unless you knocked the electron completely out of the potential it was sitting in when you fired your photon, you didn't really change the probability distribution of the electron.
Wavefunction collapse is a term that describes the particular moment of the measurement.  Up until that point, you just have a probability distribution.  I recall that some of my textbooks and some research papers refer to this probability distribution with the term: probability density function.  This is precisely what you calculate when you square the wavefunction.
If you take the view that nothing is real until it is observed, then you aren't wrong.  But in all practicality, we don't live in a world where we accurately perceive events that occur on a timescale measured in Planck time.  And you will likely have problems understanding the concepts of modern quantum mechanics.
On to your second question:

Also, are these quantum experiments anomalies, and not how nature works, or do these quantum experiments, like double slit and quantum erasers, actually reflect how nature works even in the absence of human manipulations.

These experiments are actually how nature works.  However, as I alluded in my answer to your first question, it depends on your view.  And nature works in much more complicated ways than we see in experiments or the relatively simple calculations done in say, graduate-level coursework.  In the non-experiment / non-theory scenarios where the majority of quantum mechanics takes place in the universe, there are MANY interactions happening simultaneously.  For any given system of particles' interactions, you are easily looking at orders of magnitude more complexity than anyone has successfully calculated to this day.  We only have a limited computational/analytical ability, and that's only for the most simple systems (like the hydrogen atom, or some simple molecules).  Much can be done with simulating or calculating things for molecules using electron potentials and a variety of assumptions, but you are not looking at wavefunctions for these molecules.  So... the way that quantum mechanics is taught today gives most folks a very simple, narrow view.
A: In non relativistic quantum mechanics you have a partial differential equation like the Schrödinger equation. The solutions are called wavefunctions. It is similar to Maxwell's equations in that if you provide the values at one time you can get the values at a later time.
Unlike Maxwell the values you need to provide are complex and unlike Maxwell you need to provide them not for every point in physical space but for a space that has 3n components where n is the number of particles, specifying all those numbers is enough to tell you where every single particle is so they are coordinates for an entire configuration of all n particles. 
OK. So for every configuration we specify a complex number. The value of the complex numbers evolves in time according to the Schrödinger equation.
If you have two particles then it is possible that a wavefunction $\Psi(x,y,z,X,Y,Z,t)$ could be written as a product of two other functions $$\Psi(x,y,z,X,Y,Z,t)=A(x,y,z,t)B(X,Y,Z,t).$$ If so it is factorizable. Measurements are related to these factorizable states.
Sometimes there are operations that if you do them twice you get the same result, for instance you can make a device that splits a beam leftwards and rightwards but such that if you have two such machines then the beam that went left in the first one always goes left in the second machine and the beam that went right in the first machine always goes right in the second machine. Such states are eigen to the beams, and the device always acts on these eigen things in the same way. Measurements are related to these devices and states.
We now have the ingredients to answer your question. You can always write your wavefunction as a sum of factorizable states, each of which is eigen to some devices and those will act in totally objective ways for those devices. However if you are eigen to one device you will not be eigen to other devices so those other devices will evolve you to split into a sum each of the sums being eigen to that device.
When someone says potential they just mean not eigen to that device. And what the evolution equations says is that it splits into a sum of eigen states. Technically it was already a sum of eigenstates so by splitting I mean The different eigenstates when factorized eitha different state for other particles has the other particle wave separate.
For instance $$\Psi(x,y,z,X,Y,Z,0)=\left(5A(x,y,z,0)+2a(x,y,z,0)\right)B(X,Y,Z,0)$$ where A and a are eigen, evolves to $$\Psi(x,y,z,X,Y,Z,0) = \left(5A(x,y,z,t)C(X,Y,Z,t)\right) + \left(2a(x,y,z,t)D(X,Y,Z,t)\right)$$ where C and D are disjoint (separated).
That other particle the one that separates if coupled to different eigenstates is effectively measuring the first one. If it is a whole bunch that are required to separate the eigenstate of the first one then they together measure the first one. To measure it they also basically have to not be affected by everything else. You can't measure one thing if everything else is distracting you and affecting what you do, because then the other things could make you not separate.
Notice the 5 and 2, sometimes a term in the sum is smaller than another term in the sum and then it can have less of an effect on future evolution when both overlap. 
In our example the whole thing won't overlap again if the separated parts (C and D) never overlap again. And non overlapping things act as if they were the only thing in the world. So the things that measure it act as if only that one result exists. Since they act like they are the only thing that's what we call a collapse, it takes time to fully separate but once separated they each act as if only they exist. 
Then something else can measure the thing that measured it and so on each then acting as if there were only the thing eigen to what they measured. And since bring eigen to a device is repeatable, this opinion appears objective to the thing that measured it since doing it again gives the same results.
This is how a result can seem objective to a part (like a C or a D).
Since we've never seen a deviation from the wave evolution it is reasonable to say this is how nature is. However obviously the devices changed things, they separated the waves. A different device would have separated $\Psi$ into $3K(x,y,z,t)+4k(x,y,z,t)$ instead of $5A(x,y,z,t)+2a(x,y,z,t)$ so measurements do something. And they happen over time so the order you do them can matter, maybe you were eigen to A/a but you did a K/k measurement which makes it separate into things that are K/k so then if you do an A/a measurement you'll have to split K into A/a and split k into A/a where now you get some A and some a whereas if you did your A/a measurement first you would have gotten A for sure (or gotten a  for sure because it was already eigen to A/a so it was one (e.g. A) or the other (e.g. a)).
So I haven't really mentioned those scalars the 2,3,4,5 that appeared. When you write $\Psi$ as a combination of eigenstates you are mathematically forced to have constants there. They matter for making something eigen to one device out of things that are eigen to something else but only if they aren't separated.
In particular if you look at a device that measures how many times a bunch of things got deflected left then those constants will affect what that device is eigen to hence what we see when we measure how many of them were sent left. So they affect what the measuring device calls the frequency of observations.
So the waves appear to always follow the equation both when we are measuring and when we are not. Measuring does have an effect. And the order in which you measure changes your results. But we never measure the numerical values of the wave all we can do is evolve it so that it becomes parts that then act independently and so each part can think it is the whole thing. And that appearance appears objective because if that part does the same measurement twice it gets the same result both times (unless it does a different measurement in between since measurements split things that aren't eigen to it so changes them).
