I remove my hop bag a minute before I turn off the burner and squeeze the wort out of the bag by pushing the bag against the side of the kettle with my brewing spoon. Doing it a little before you cool it off will disinfect any nasties that I may have introduced with my spoon.
I pull mine, but due to my setup, it would be a pain to chill with the bag in. Still trying to find a good way to get my IC into the boil for 10 with a strainer bag stretched around the rim of the pot. :/
Really depends on what I'm brewing. If its a hop forward type of IPA or pale, i will leave it in while cooling. It it's not a hop forward type of brew, I take it out at flame out. I use a hop spider for the pellet/leaf hops, fwiw.