It has many advantages, and only one gotcha, that being that a legitimate network partition could get into a state where it is not possible to reconcile the two halves manually.
Er. More than one. You end up with a byzantine generals problem in determining that a situation exists which requires the higher difficulty. Normally we solve the byzantine generals problem by using the blockchain, but in this case we'd need to secure the blockchain a blockchain and we'd have infinite recursion.
Nah. It is just counting, and the decision happens on each node.
For what it's worth, I think that a permanent fork would be close enough to impossible not to worry about. I can't think of any way to divide a large fraction of the world's hashing power from the remainder for more than a couple of hours. Well, I can think of a one way, but if alien spaceships shoot down all of our satellites, jam all radio and microwave transmissions, and cut the earth in half, we'll have bigger things to worry about that day.