Has it been proven (or disproven) that, from a solved state, there exists a number “x” for any algorithm, such that you can perform said algorithm x times to get back to a solved state. I feel like this is something any cuber will tell you is true, but I am a mathematician first and a cuber second. I want to prove (or disprove) this mathematically if it hasn’t been done.
The question is a bit unclear - it sounds like you're asking "Given a sequence of moves A, is there always some number of repetitions of A that will get you back to the start state?"
This is relatively easy to prove - it's just basic group theory.
Imagine you have a bunch of Rubik's cubes -- one in every possible state. Given an algorithm A, you can draw a sequence of arrows from one cube to the next: if you start at cube 1, apply A, and end at cube 2, then you draw an arrow from cube 1 to cube 2.
Every cube has only one arrow pointing out of it. Similarly, since you can just "undo" A, every cube can only have one arrow pointing into it.
So, if you start at the solved cube, and keep following the chain, what happens? There are a finite number of cubes, so you can't keep going forever. And you can't first repeat in the middle of your chain, because then the repeated cube would have two different cubes pointing to it. So you must hit the solved state again.
If "algorithm" does not mean "sequence of moves, independent of the current state of the cube", then the answer is easily 'no'. The algorithm "Always turn the right face, unless that would solve the cube - then turn the left face instead" will never solve the cube, no matter how many times you apply it.