I don't use Eagle, so I cannot tell you how to change the min drill size, but I would say to make those mount holes a whole lot bigger. Right now, they are pinholes.
As far as the capacitor, unroute the traces, move the capacitor, and place new traces in. I would get those traces a big bigger, and get them away from other pins. The traces at pin 3 and 5 have no reason to be that close to the pins. Just a little stray solder will short them, and I'm not sure the soldermask will adequately protect them.
The ground PLANE can be on both sides of the board, and will actually replace all the ground traces. The whole surface will be covered, except for the existing traces and a small area around them, which you can set. ^The ground plane will connect with the negative terminal of the battery, the ground pin of the ISP, the IC, the piezo, and anywhere else you have a ground connection.