Electrical – does adding Vias through the antenna feed line change impedance

It looks pretty reasonable to me. I've got a question and a comment.

What are the components between the antenna and the GPS module? Any clocks etc? If they are power supply lines, it looks a little light on bypassing.

I'd also drop a dozen or so ground vias under the antenna itself, placed randomly. The reasoning behind this is to make sure you don't create a resonant cavity under the antenna, with the walls formed by ground planes top and bottom, as well as the via holes running around the outside. Unlikely, as your pcb is likely to be lossy, but you'd hate to hit the jackpot with that.

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.