Is every composite number the average of two primes?

So you are asking whether for every composite number $n$ there exist primes $p,q$ such that $$n=\frac{p+q}{2}$$ That is, $2n=p+q$, so you are asking whether $2n$ can be written as the sum of two primes.

The question whether every even integer greater than $2$ is a sum of two primes is a famous open problem known as the Goldbach conjecture.

This is almost the same as the Goldbach conjecture, that every even number from four is the sum of two primes.
It has been checked into the quintillions (no, really!) but not proven. In 2013, a similar theorem was proven by Harald Helfgott, that every odd number from seven up is the sum of three primes.

A generalized equation would be in the form of:

$$x =\frac{P_1+P_2}{2}$$

Where $x \in Z^+$, and $P_1,P_2 \in Z^p$ (Here, I define $Z^p$ to be a prime number).

It can be rewritten as $$2x = P_1+P_2$$

Let $2x=a$ Then the equation is: $$a = P_1+P_2$$

I.e. Your question's generalised form ($x =\frac{P_1+P_2}{2}$) is akin to the Goldbach Conjecture: http://en.wikipedia.org/wiki/Goldbach%27s_conjecture