When has the Borel-Cantelli heuristic been wrong? How about the Maier phenomenon that occasionally there are intervals around $x$ of length $(\log x)^{100}$ say with substantially more (or fewer) primes than one would expect?

distribution of $\sqrt{-1} \mod p$ Hooley proved the equidistribution of roots $\mod d$ for composite $d$. That is a much easier problem than the one resolved by Duke, Friedlander and Iwaniec.

distribution of $\sqrt{-1} \mod p$ I think they want the determinant to be positive; ie the quadratic polynomial has complex roots as in $x^2+1$. Also I don't see how uniform distribution of the angle helps, since we need the distribution of $ab^{-1} \pmod p$.