1) I am not an "antenna guy", so take anything I say with a grain of salt
2) As I understand patch antennas, the radiation is from the 4 edges, where the Efield is strong from the patch to the underlying ground plane. To get the Efield to be a maximum, you need the width and length of the patch to "resonate". You can lower the resonant frequency of the patch by capacitively loading the edges (makes the width and length appear physically longer). There is no need to put the capacitors on one edge and the opposite edge, as you can get the same electrical length with the capacitors on only one edge. So I would pick one edge, and one of its adjacent edges, and load them with variable capacitors. You might be able to do it with one varactor right in the middle of the edge, but I suspect it would work better (more uniform radiation pattern) if you had multiple varactors spread along each of the 2 edges.
3) I do not understand what the inductor's purpose is in the "full patch" drawing above, so I can not effectively comment on it. If the inductor is used to tune the patch, then yes, a variable capacitor across the inductor will vary the frequency of resonance also. I am not sure if you can get away with only having variable tuning on one edge of the patch. It seems that if it did work, you would not be able to tune very far.