Is there a general process that I can follow to find the period of a function or to prove that there isn't one?
Everything I find on the matter has something to do with trigonometric functions and finding LCM of the periods.
Are there continuous periodic functions that are not trigonometric?
Are there differentiable periodic functions that are not trigonometric?
There definitely are periodic continuous functions that are not trigonometric: I can take any continuous function on an interval, say $[-1, 1]$ and then repeat it periodically. In this case though, we can actually expand this periodic function into a Fourier series, which is a sum of sine and cosine functions. So in some way, trigonometric functions are the only continuous periodic functions.
Similarly, if you are periodic and differentiable, then you are continuous, and so the above discussion also holds in this case.