Have run many an early morning run with a hangover. Don't like drinking and then going running that much but have done that as well. Usually been out drinking the night before until midnight or 1AM. Get home, sleep, then get up around 6:00am and head out the door for a 5 mile recovery run. It definitely gets last nights festivities out of my system and ready for the day ahead.
When I worked overseas I used to play softball with a group of expat guys at a local school on Sunday afternoons. We always played a double header with a 20-30 minute break in between. During the break we would have something to eat and a couple of beers. We always had a two beer limit. You couldn't take the field for game 2 unless you had drunk two beers.
Personally I feel super sluggish when running after drinking. I don't think I run faster, and I definitely don't feel super good. Running is definitely one of the last things I want to do after drinking, unless it's a beer mile.