Fibonacci, Four Ways

The sequence $\{a_n\}$ satisfies $a_1 = a_2 = 1$ and $a_{n+2} = a_{n+1} + a_n$. Determine, with proof, which of the following are true:

1. only finitely many positive integers $m$ have $a_m$ and $a_{m+1}$ coprime; 2. there exists at least one positive integer $m$ with $a_{m+2}a_m = a_{m+1}^2$; 3. there exist infinitely many positive integers $p, q$ such that $a_pa_q - 1$ is a perfect square; 4. there exist infinitely many pairs $(m, n)$ of positive integers with $a_m \mid \left(a_n^2 + 1\right)$ and $a_n \mid \left(a_m^2 + 1\right)$.