Ok, let me address these one at a time:

What's the difference between sin^2(x), (sinx)^2 and sin(x)^2?

I'm not exactly sure what you're saying, we CAN say that:Why is it that we can simplify some trig functions like

sin2x/2x into siny/y, but when we put a radian value in for x in the first expression it doesn't equal to the same this when we put that value in for y? I think this is using stunt variable but I'm unsure why we can use these things are true equations to find other answers (i hope this question made a little bit of sense)?

BECAUSE x is in radians, it must be dimensionless to do so, real numbers have no units, radians have no units. In this case, y MUST equal 2x, or whatever you choose to put into the sine function instead of 2x. y is a dummy variable to make manipulation easier, or to make it easier to understand.

There is no difference except maybe what the function actually is. There are many labels for functions, f(x), g(x), h(x), I(x), etc. The names may mean something, or may not mean anything more than a function of x. If it means anything at all, it will be specified in the problem you are working on or in the course you are in.Also what's the difference between F(x) and f(x)?