Much of the frequency variation comes for the varactor side. Junction caps can modulate the  frequency as well but with a much smaller gain. Therefore, varactor biasing needs to be designer very carefully and might very well require large on-chip capacitance.

As for the voltage-biased versus current-biased, the answer is yes, current biased VCOs are better wrt to pushing and this is predominantly because the exhibit less AM noise with supply variation. This, of course, if you bias the current source from a clean reference.

hmm, depends on the values of the devices in the tank, core, Ctail, varactor/FET CV curves etc.. what i would do in this situation is simulate to get the answer. you can check how much a signal on ground or VDD pushes/pulls the VCO with both ground and vdd referenced loop filter. assuming noise on vdd or ground is small, using something like spectreRF pxf could be appropriate.

Typical a VCO have 3 supply ports.

VDD, VSS and SUB

The VDD could be regulated. All 3 are AM/FM sensitive. The nonlinear diffusion cap typical is the source of the conversion. Some tricks are to have a second set of diodes where the nonlinearity is stronger but the absolute cap contribution is lower. Then this set of second diodes are connected to other supplies. So if a supply voltage change the cap of the first set goes up and the cap of the second set goes down.