Hi, I have 2 coop issues that was hoping someone would have some advice. The nest box lid has a big gap, what are some suggestions to fix this and also should I let the space under the roof open for
You should be able to take the hinge screws out of the bottom piece and just slide it up a bit, and rescrew (assuming there's a gap built into the board below so it can pivot).
As to the top, I'd fill it with hardware cloth. Ventilation is almost always a good thing.