I'm not a mentor for our team (until next year!) but i can answer a number of your questions
1. Our students design and build the robot in the school's technology lab. We have one mentor who is a master machinist and will do our very custom parts on a CNC mill, he also helps with build and design when we're stumped (our other mentors do this as well) but the majority of the robot is student done. Looking at our robot this year, you'd be surprised how many of the parts were hand made by students.
2. We have a small room that contains a hand mill and lathe, but the school also has a room with two bansaws, a drill press, sanders, grinders, and all manner of other tools. There are also classes in the room that we build in, so we have to clean up everyday when we finish working.
3. Do you mean corp. sponsors to build for us? or machine special parts? it's a fairly ambiguous question.. we have certain special parts made, but the majority of the stuff is student done and run.
4. We have a combination of student leaders and mentors that keep everybody working and having fun. But the mentors also try to encourage the students to be a little bit self run.
5. Our school is not very involved at all. We have to have one teacher sponsor from the school (basically to say we're with the school) but other than that we're more or less unknown by the student population. The principal, surprisingly enough, has begun to show immense interest in the team and our well being.
6. Our student parents are kind of involved. We have one that is extremely involved, one set that are always willing to drive students and our trailer to competitions, and then a few who volunteer saturday lunch duty during build season. Overall, we don't have much parent involvement, but we're always welcome to new parents and mentors.
7. Our operating budget depends on how much we get from our sponsors year to year. This year we've been fairly lucky, having enough money to go to three events and still keep things cheap for the students. But i'd say most teams operate on $10,000 or more (depending on the number of events and so on)
8. None of our mentors get paid by their companies or by us, we don't require them to do anything for us, they choose to come so that they can help our future, and be a part of something good (and tons of fun, they're all kids at heart you know
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9. Our typical build season schedule is meeting after school (2:10) on mondays through thursdays until around 6-9 (depending on where we are in build season) and then 9-5 on saturdays. On the last week of build season we also come in on sunday, and this year we also came friday. Also, whenever we have mondays off from school for teacher workdays, or student holidays (during build) we come in and stay 9-5. All of our times are also flexible, but we don't require anyone to be there at certain times or else, and we don't hold people from leaving.
If you have any more questions feel free to contact me, we are a seven year team, located near the atlanta area. I can also put you in contact with a number of long time mentors and FIRST volunteers. Between them, i'm sure you could get all the information (and more) that you might ever need