The KV rating indicates an unloaded ( no prop) RPM that the motor will spin at. The actual RPM you get will depend on the prop you use. The ESC will work with a Gens ace 4s lipo battery, but for any given set up ( load and throttle setting) the 4s will result in a higher average current delivery than the 3S.
The 11.1 v assumption you are making on the the 3S 11.1v lipo battery battery is the low end of the charge/ discharge voltage curve, I would use a fully charged battery voltage to think about current draw. If the top end the motor will deal with is 660 watts, at 12.5 v that will be a current draw of 52 to 53 amps, which exceeds the capacity of your ESC ( 45 amp continuous, 50 burst). The ESC seems to be the limiting factor in your power system, so to use the 4S battery you may need to consider reducing your prop size and or pitch to maintain the same overall current draw on the system.
Because the ESC deliver current in a time gated fashion ( full current on then off with the interval determined by your thottle setting) you cannot depend on the just assuming you can fly at less than full throttle to keep your current delivery in a safe range for your ESC.
There is the benefit of getting more watts out of your setup. The Afro 20A slim ESC (2S-4S) and 18AWG wire are rated for 20A. Amps is what melts your wires and ESCs. Watts is what makes your motors powerful generally speaking.
For each motor:
A 3S setup will have a theoretical watt limit of 12v x 20A = 240 watts
A 4S setup will have a theoretical watt limit of 15v x 20A = 300 watts
Also the LiPos “contain” more watts per mAh in a 4S lipo vs a 3S lipo:
Lets compare 2200mAh 40C lipos
2200mAh 3S 40C lipo: 2200mAh x 12v = 264 watts – continuous draw: 88A
2200mAh 4S 40C lipo: 2200mAh x 15v = 330 watts – continuous draw: 88A
4S tends to be a bit heavier than 3S with the same mAh and C rating though. What makes most sense is to compare the watt to weight ratio.
Another vital factor to be aware of with the Silver series ESC is that they use a % of initial voltage to determine the LVC, as opposed to the fixed voltage that most other manufactures use. As stated in their instructions, never use a partially discharged battery on thier ESC as it may set the LVC at a point that will result in battery damage if flown to the LVC.
The motor will WANT to increase speed by 1000 rpm (per volt), but will lose some to heat and what have you. I wouldn't get too bogged down on that part right now.
Watts = volts X amps
A higher VOLTAGE setup is often used to REDUCE the AMPS required. If you look at the formula above, if you increase the voltage, you reduce the amps.
The esc amp rating has NOTHING to do with the amount of power your setup will draw. It is a MAXIMUM rating for the esc. Think of an electrical fuse. You can flow anything from zero up to the fuse limit with no problems. Exceed the maximum rating and bad things happen. Smaller esc's are usually used to save weight, room, and money.Similar with the C rating of the battery pack. C rating x the battery capacity (in amp hours) = the number of amps the battery CAN provide at its maximum level. You can certainly use a higher C rating battery, but not a LOWER C rating.