do even functions always have even exponents? (vice versa)?

This is correct. If we restrict ourselves to polynomials, even functions have only even powers of x and odd functions have only odd powers of x - this is in fact why we call these functions "even" or "odd".

At the root of it is the fact that (-x)^n = (-1)^n x^n which is equal to -x^n if n is odd, or x^n if n is even.

So a function made up of terms of the form cx^n (i.e. a polynomial) will be even if all the n are even, and odd if all the n are odd. It's possible but somewhat harder to prove that the converse applies - that if a polynomial is even, all its nonzero terms must have even powers, and if it is odd, all its nonzero terms must have odd powers. (Hence, as you guessed, polynomials with both even and odd powers are neither even nor odd functions.)

If we're not talking about polynomials, this rule doesn't apply - for instance the absolute value function |x| is even, but x only appears as an odd power.

I've wondered about this for a while, and I'm guessing the answer is yes?

Because even functions give the same output for any number and its negative version, that's their definition. So if f(x) is even, then f(5.2) = f(-5.2).

Here, all those even exponents are like squares of something, and squaring something (whether it's positive or negative) will always give the same positive square.

That's why even functions are symmetric around the y-axis.

Same goes for odd functions, kind of.

Any odd powers of x will be negative when x is, like x³.

So if f(x) is odd, f( -x ) = - f(x).

And functions with mixed exponents would be neither, obviously.