This seemed simple, but I keep getting ridiculous fractions.

For example, find the GCD of f(x) and f'(x) over Q of:

f(x)=x^3 -3x -2

The quotient becomes 1/3x with a remainder of -2x-2, which can be simplified to x+1.

From my understanding, x+1 is the GCD. This doesnt seem correct,

But it is: just check that both have -1 as one of their roots.
taking the quotient into consideration. Can anyone check this?

Also, as a very similar question, if I get a remainder of -2x^2+x, can I simplify this to -2x+1, or does such a simplification not work for GCD's?

What do you mean "simplify it"? You're dividing it by a non-trivial divisor! That's not simplifying but changing. Perhpas what you mean is that, for instance, are associate or equivalent elements since one equals the other times a unit. In your case, if is the gcd, then you can "factor out" that unit and say the gcd is Tonio