[popnbrown]

A few teams I know about are primarily based out of One school but invite other schools within their district.

You have two options here, and they're both hard but it depends on the situation:
A) is there any other faculty or staff that can liaison with the school? There are several teams I know including mine where the lead mentor is not an employee at the school but there is A mentor that is and serves as the administrative person for the team (regarding finances, facilities)
B) are you capable/willing of setting up a 501(c)(3)? This means are there also other adults who would are invested enough to help you?

My honest opinion is that A sounds like a much better option for this coming year than B. And it'll allow you to eventually get B up and running so that if any ties must be broken with the school you can still run the team independently.

To answer your questions:
A. That's a loaded question but a 501(c)(3) really helps and a ton of community support. Especially parental support. What specifically are you wondering About?
B. if you are the main contact in TIMS, then there's no need for a new team #. if you are not, then it's a matter of figuring out how you work it out with the school. I would suggest sticking to your team #.