How long should a random id be to avoid collisions

Use the birthday approximation: collisions become likely around \(\sqrt{N}\) draws from a space of size \(N\). For a 64-bit id that is about 4 billion items, for 128 bits it is beyond anything you will generate. The practical answer for most systems is 128 random bits and no coordination.

No, and the gap is the whole point of the rule:

With a prevalence \(P(D) = 0.001\) and both error rates at 1%, this gives roughly \(0.09\). Nine percent, not ninety-nine, because false positives are drawn from a pool a thousand times larger.