I use a PCB mill to make my boards, and it doesn't etch off excess copper. I'm designing a breakout board for an RF tuner (400MHz), and I planned to have the ground plane on the top and the traces on the bottom. The traces will be surrounded by copper that isn't part of the circuit, just left over from the milling process. Is it ok to connect this top ground plane (left over copper) to the bottom ground plane, to help noise reduction? Or is there a problem with having ground planes on the top and bottom?
Yes, tie them together. Do NOT leave any floating copper since it can form an unintentional patch antenna. Note that grounded copper on the trace layer may lower the trace impedance slightly.
You should add plenty of stitch vias to make sure the copper islands on trace layer are grounded to the plane with low impedance at 400 MHz (that frequency was chosen because you mentioned it in your question). Any islands which cannot be stitched to the main GND plane with vias should be milled out altogether or removed with hand tools.