I rarely cut Quartz wet when in the field.

If you're drilling a quartz counter top wear a respirator and use a vacuum. Those tops are 93% Silica and 7% resin. Highly toxic. I'm our stone shops Safety guy!

If you want to be really safe remove the faucet and measure front and back of the existing hole and you can find center. The existing holes on the sink should be 4" on center.

Yes. Some core bits have a center bit. We have diamond coated router bits that we use with a large water feed to keep the bits cool.

The problem with coming up from the bottom is that you can blow the face out. If it's a marble or granite type surface it wouldn't be able to handle the heat and have fun trying to use water to keep the bit cool while working underneath the sink.

Stone guy here. I mount sinks for a living. The pre drilled holes in the flange re 4" on center. If you found center of your sink to the right and left you should have 2 holes 4" apart from each other. There should be five holes in the sink all together all 1.375 - 1.5 inches in diameter. Drilling through the stainless can be done but it's not optimal. If you overheat the bit the steel will bruise and turn blue. You can drill a larger hole depending on the size of the escutcheon plate. What kind of material are you drilling?