In RSA, must $p$ and $q$ have the same number of bits?
The RSA method works regardless of the size of the individual primes $p,q$. The reason that we would want $p$ and $q$ to be of roughly equal size is that one method of breaking RSA is factoring $n = pq$, and the running time of some factoring methods (notably ECM, the elliptic curve method) depends on the size of the smallest prime factor of $n$. Therefore to get the most "bang for the buck", you would like $p,q$ to be both as large as possible under the constraint on the size of $n$ (which determines the speed of encryption and decryption), and so of roughly equal size.