I'm a drummer in a band; is it common for drummers to wear headphones and to listen to the metronome while playing with a real band?
Good question! None of the bands I've worked with had drummers that used metronomes while playing live - but most are good timekeepers anyway.Except one where we used backing tracks, but he was at the back and had an uncanny sense of timing, so didn't need more than the count-in.
If a recording is going on, then usually the drummer will use one. It's the click track that everyone needs to keep time to. If there's pre-recorded tracks that get played along with the band at gigs, then yes, it's important the drummer can hear exactly what the timing is.
In itself, it's a great idea to keep everyone together - which is one of the main jobs a drummer has. So listening to one whilst playing can help a lot. Providing, of course that the songs played are using a constant beat!
If you're asking on your behalf, then yes, at least in rehearsals, and also in selected gigs. If you're asking because the drummer in your band wanders time-wise, it may/may not go down too well. Most drummers think they are human metronomes..! But if he agrees, it may be even better to use a click that everyone can play to together - sometimes it's not the drummer who skips timings!
Yes, definitely. There might be a "click track" or some sequenced backing that the live drummer plays along with. Just about every pro drummer in the pop world does this at some point. (If you look online for an old edition of BBC "RockSchool" with Omar Hakim, he did a very impressive demonstration of changing the feel by playing slightly ahead of or behind the sequenced track.)
(They often use in ear monitors which you cannot see unless you are very close up, though.)