The general proposition of Fermat
Let $p=a^2+2$ be a prime. Suppose $a>0$ as for $a=0$, the statement clearly does not hold. Therefore, $p\equiv 3 \pmod 8$. Thus, the congruence $x^2\equiv 2\pmod p$ has no solutions$($$2$ is a quadratic residue modulo $p$ if and only if $p\equiv ±1\pmod 8$$)$.
Ad 3): If $p = a^2 + 2$ divides $x^2 - 2$, then $p$ divides $x^2 - 2 + p = x^2 + a^2$. But primes $p = 4n+3$ cannot divide sums of two squares without dividing the squares themselves. This observation is due to Weil.