If the network was split fairly evenly, both sides would immediate notice block times roughly doubling due to hashrate halving until we hit the next retarget. Outside of this, though, I'm certain that everyone would immediately notice the internet splitting in two, since you would not be able to communicate with many of your contacts and you would lose access to huge chunks of content.

Provided neither chain rolled out any update or fork which would make it incompatible with the other chain, then when the connection was reestablished whichever chain had the most proof of work would win and the other chain would simply be abandoned. I would assume in such a doomsday event though that at least one side would deliberately take steps to prevent this from happening so as not to risk a massive reorg and loss of hundreds or even thousands of blocks worth of block rewards.

Not as far as I am aware, and with nodes being run over Tor/VPN/VPS, then IP addresses are not reliable indicators anyway.