Apparently, his coworkers call him Bing. (I never watched Friends, so I don't know.)
When he says "I ask myself the same question" he's replying that he doesn't know why a lady as pretty as Monica would want to be with him either.
Law firm - A company that provides legal services.
Drink our body weight - if you weight 60 kilos, then 60 kilos of liquor would be a LOT. It's obviously an exaggeration - he's saying "let's go get drunk"
A work laugh - apparently, he has a different personality at work, so his "usual" laugh would be out of place at this office party, where he work friends know him. And Monica should come up with a "work laugh" as well. Probably, it's a more polite, low-key laugh than a good, hearty laugh like you would use if your friends made you laugh.
Check me out - look at/listen to me.
I have no idea what the feathers thing is about.