1. I've had, approached and been approached by sponsors in the past, and it varies depending on them. A lot of gaming communities, as mentioned, sometimes sponsor competitive teams for their games.
2. Depends on the sponsor. I think it's perfectly fair for them to ask you to change your name, or at least your tag.
3. Again, this depends. One sponsor I had wanted us to post on their forums consistently, which was weird and we didn't bother with them. Others may ask you to provide something on your team page that links back to them and their sponsors. Others might want you to say something about them in matches or something; usually it will be something reasonable, and worth getting a free server/fees/etc.