First of all, I wouldn't expect anything to be provable, but I expect you can get a good guess for the first question by using Granville's modification of Cramér's random model for prime distributions.

For the second question, assume that $p_{n+1}-p_{n-1}<\sqrt{n}/10$ (this is conjectured true for all sufficiently large $n$, but even just the prime number theorem says this is true of all but $O(\sqrt{N}\log^2 N)$ primes less than $N$).

Then
$$
\frac{p_{n-1}+p_{n+1}}{2}-\sqrt{p_{n-1}p_{n+1}}<0.5
$$
so the question is asking how often $p_{n-1},p_n,p_{n+1}$ are in an arithmetic progression. The difference would have to be a multiple of 6 for all three numbers to be prime. You can get a good guess by summing over all possible multiples of six and using the Hardy-Littlewood conjecture.