According to a quick search, you can rent out the amphitheatre at the Mill pond for a full or half day. I would also suggest following the advice of other commenters and sending a quick email to city council just to double check.
The City's DPS rents out the space. 810.225.8001
They actually just recently renovated the shelter and it is very nice. I'm sure that you can reserve it, there was a notice of sorts there a little while ago.
You would have to ask the City Council and get a permit.
Not sure, but you would likely have to talk to someone about reserving a space. It is a public space so there is no guarantee that someone else may not be occupying it. Probably check with town hall and see if reservations are something that can be done
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