Answering for 2177...
Who builds robots/where?
It's an equal combination of mentors and students. Generally speaking, the mentors do almost nothing alone, while some of the veteran students can take tasks and work autonomously (sometimes for entire meetings). Our build space is located off-campus, at a place the school started renting for us. We ended up there mostly because of a machine shop that sponsored the team for several years worked in the same building.
Do you have dedicated space/machine shop?
yes, off-campus. This is the first year it's really been dedicated, however. Previously, we've had to move around to a different location/room in the building each year. The school signed a 3-year lease and renovated the space for our needs, and we officially moved in the week before kickoff.
Do you rely on corp. sponsors to build?
Yes and no. Boston Scientific donates a significant amount to the team, as do other corporate sponsors. Additionally, the teams 4 mentors all work there and became involved with the team through the company. Further, the machine shop at Boston Scientific donates some time every year when we have parts that are beyond the abilities of the equipment the team owns.
Who manages the team/students?
There are 4 professional engineers serving as mentors with the team, and 3 teachers from the school. Managing the team is really a group effort, however one of the teachers is the official adviser for the team and does more work as such, and one of the engineers works as a "lead mentor", essentially responsible for ensuring building the robot happens smoothly.
How involved is your school?
More involved every year. As already mentioned, the school has signed a 3-year lease on a build space for the team. Additionally, a major school donor helped fund the team when it was starting up, and the school has also helped donate money to purchase equipment. 3 teachers from the school work with the team this year (last year there were 2, the year before that 1).
How involved are your students parents?
Every Saturday the parents bring in lunch for the students. Some of the parents also donate time with the team on specific, non-robot related tasks - mascot costume, building a practice field.
What is your teams operating budget?
too much

We attend two regionals every year and put an extra ~$1000 worth of parts on the robot that aren't included in the KoP. One of the regionals is in town, so we don't have to pay for any travel, but the other is out of town. Bus + hotel rooms for 30 people adds up quickly. Plus we try to set some aside every year to buy more equipment. If the school doesn't have any equipment already, you can expect to spend $500-$1000 right off the bat gearing up the team. If your regional is in town, you would only really need the price of the KoP and regional, plus some extra for more parts.
How do you get compensated, besides the obvious joy of watching your students grow?
I get to put it down as one of my performance objectives for the year, which can help give me a little extra boost when compared to my coworkers. I believe the team adviser from the school gets a small additional stipend.
What is your typical build season schedule?
Monday-Thursday 7-9 (sometimes 6-9 if needed)
Saturday 9-4
Expect that to go out the window the last week of the build season, however. The weekend before ship, we typically plan 6-midnight on Friday, 9-4 on Saturday (the school usually has a dance that week that we have to stop for. It can be great seeing students show up with curlers in their hair to work on the robot!), 3-9 Sunday and 6-whenever Monday.
One thing can't be over-emphasized for a new team with new mentors - this is a BIG commitment. It is extremely difficult to have other social engagements during the build season - the few nights you have off you'd rather get some sleep than go out and do stuff. It is incredibly fun and rewarding though, which more than makes up for it.