Patch antenna ground plane
This should be answered in the datasheets.  Without knowing anything more about your particular situation than you presented here, I would assume that there should be no copper at all below the patch antenna.  The keepout rectangle probably extends a bit all around the patch antenna too.

Again, details should be in the datasheets.  If this patch antenna is for a specific frequency or specifically for GPS receivers, then its datasheet should give you all the relevant layout rules.  There should also be rules on how the connections are routed.  Metal nearby can significantly change the antenna characteristics to the point where performance is degraded.  The GPS module datasheet may also have information on how it can be connected to particular antennas.