Bayes Theorem Example in Nate Silver's The Signal and the Noise

So we can be almost certain the second plane crash is a terrorist attack, and we can be even more certain that it's accidental?

Correct, there is no contradiction here.

If we know that the first crash was a terrorist attack, then the second crash would be more likely another terrorist attack.

The same reasoning with accidental crashes.

Question 1: Am I correct that Nate Silver is doing it wrong? Question 2: Am I doing it right?

No. There is no need to update the rate of accidental crashes. IMHO, Nate implies that accidental crashes don't include terrorist ones. Otherwise, he couldn't multiply probabilities in the denominator.

though the chance of two accidental plane crashes can multiplied to give you a very small number (1/12500 x 1/12500) since they are independent, one cannot assume the same for a terrorist attack. Once we think that the first plane crash is a TPA, it would not make sense to assume that the second crash, if it is also a TPA, to be independent and not highly correlated (perhaps 90% chance that the 2nd plane crash is TPA given the first is TPA) to the first. so if you use 1/20000 x 0.9 to get the probability that both plane crashes are TPA, you will not end up with the problem you mentioned that both scenarios have become more likely.