That's a good question, one that's come up recently in discussions about removing the new downed trees on the Shenango River.
I've removed trees from whitewater creeks for years - just went in and did it. Clearing Mosquito Creek however would be an extended task. There are likely dozens of trees down.
Some waterways have state or federal protection, or lie within state lands. Mosquito Creek isn't protected, but part of it may be within Mosquito Lake State Park land, and more of it flows through Trumbull Metroparks land.
So I don't know of any restrictions - but that doesn't mean there aren't any. You might try talking to the Mahoning River Consortium, and see if any of their staff has a legal overview of the issue.
http://www.mahoningriver.com/
BTW if you like paddling and fishing Mosquito Creek, have you tried taking Pymatuning Creek upstream from the township park in Orangeville?
Steve Z