Can we find a formula to generate all prime numbers, and is the distribution of primes truly random?
There are several formulas that generate all prime numbers, but they're not efficient enough to be of any practical use. For example, a formula devised by C. P. Willans in 1964 does technically return the $n$th prime, but it does so by summing $2^n$ values, each of which requires a factorial to compute. 

As for randomness, as the set of prime numbers is fixed, it can't be truly random, only pseudorandom. It's also not uniform, as evidenced by e.g. the Ulam spiral and its unnatural frequency of straight lines.