My second observation is the following. Let $p$ be a prime not equal to $5$. Then $5$ is a quadratic residue modulo $p$ if and only if $p\equiv\pm1\pmod5$. And $5$ is not a quadratic residue modulo $p$ if and only if $p\equiv\pm2\pmod5$.
If $p$ is a prime and $m$ the period of $F_n\pmod{p}$, then $p\equiv\pm1\pmod5$ implies $m|(p-1)$.
I am looking for a generalization of the above cited statment.