Are you allowed to fish in any of the ponds?

Yep

I believe there is a designated lake area for fishing.

Yes you have to catch and release though if you don't have a lot license or permit.

I saw a lot of signs near lake that say No fishing. They also say No pets at the lake, but guess what. There are snapping turtles in there that are getting fed by people, so I recommend don't fish or good luck getting that hook out.

Just a change of scenery and it's down the street. Is there even any fish in the ponds?

Yes.

Plastic

Yes , I don't know the real reason why you are picking through fish there may be for family but there are a lot of private fishing spots that I'm just as beautiful and private