1) is divisible by in , so no, it is not prime. making that number divisible by 31.
2) haven't thought about it
For (2), I would think it hard to prove there are only finitely many such primes. Usually,there is a proof that there are infinitely many such primes. So, I would attempt a proof by contradiction starting with the proposition that there are only finitely many primes such that and are both prime. Have you attempted this at all?
Such a prime, if it exists, must be of the form $6k + 5$. To see this, note that:
$6k$ is divisible by 6
$6k + 2$ and $6k + 4$ are divisible by 2
$6k + 3$ is divisible by 3.
That leaves only $6k + 1$ and $6k + 5$. If we have $p = 6k+1$, then $p+8 = 6k + 9$ is divisible by 3.
(this is really just saying $p$ is of the form $3n + 2$).
Or, for question 2, with the exception of p=5, try modulus 30 to weed out the 5's later on. XD
All candidates listed: 5, 11, 17, 23, 29.
Clearly 11+14=25, 17+8=25, and 5|25, so the first three can be weeded out.
Now, check p if p=23 or 29 (mod 30)