Some heuristics about the Pisano Period, primes and Fibonacci primes. What reasons are behind them?

I think first three statements are all false. Numbers with these properties are analogous to Fermat pseudoprimes, and in particular there's no reason to expect that they should in fact always be prime, although counterexamples might be quite large.

Using Binet's formula as in Jyrki's comments, you can prove results like the following. Let $p \neq 5$ be a prime. We will need the Legendre symbol $\left( \frac{5}{p} \right)$, which is equal to $1$ if $p \equiv 1, 4 \bmod 5$ and $-1$ if $p \equiv 2, 3 \bmod 5$. For reasons that will become apparent I'll write $F_n$ as $F(n)$.

First,

$$F(p) \equiv \left( \frac{5}{p} \right) \bmod p.$$

Next,

$$F \left( p - \left( \frac{5}{p} \right) \right) \equiv 0 \bmod p.$$

These are the two basic results, analogous to Fermat's little theorem. Together they allow you to bound the Pisano period of primes as follows: if $\left( \frac{5}{p} \right) = 1$, then the Pisano period divides $p - 1$. If $\left( \frac{5}{p} \right) = -1$, then the Pisano period divides $2(p + 1)$.

This is a partial explanation of your first observation. For your second two observations we have the following slightly harder result. If $p \equiv 1 \bmod 4$, then

$$F \left( \frac{p - \left( \frac{5}{p} \right)}{2} \right) \equiv 0 \bmod p.$$