To get E(f(x)) where x is uniformly distributed between 0 and A, you simply compute:
1/A integral from 0 to A of f(x). In your case A=pi and x=theta. This will give you the answers you need for (a).
The answer to (b) depends on the answer to the middle expectation. It will be a function of k and m. It is wide sense stationary if it is a function only of k-m, and not of them individuallly.