It is easy to lose track of the Riemann Hypothesis behind all the noise about the Riemann Hypothesis. But following the usual statement of the problem (where I use the Clay Mathematics Institute and their prize as the usual statement), the hypothesis is that all the zeroes of the Riemann Zeta function occur on the line $\Re(s) = 1/2$.
With respect to your first question, the Riemann Zeta Function is traditionally used in one proof of the Prime Number Theorem (which states the asymptotic density of primes). In this way, it gives that there are infinitely many primes.
Alternatively, as pointed out by N.S. in the comments, the fact that $\zeta(1) = \infty$ is a direct statement that there are infinitely many primes. And 'calculating' this is not so bad, as long as you know that the harmonic series diverges and that $\sum\limits \frac{1}{n} = \prod_p \frac{1}{1 - p^{-1}}$.