first, use stainless steel screws and put one in every 8 inches. You cannot over screw the floor down, matter of fact the more secure the floor the less problems you will have later on. Majority of the screws also need to hit the floor joists. A slight movement or bounce in the sub will crack the grout in a minute.
Also, make sure that the sub is thick enough.
Perfect circle, well you can get a circle cutter that mounts to your drill. Most of the circles that I cut are crude because a decorative pipe cap fits over the fitting to hide the actual circle,
There is also a ceramic blade that you can attach to a hax saw that cuts a pretty good cut. Nippers in the hand of a patient person will also work. There are many ways of doing this, just depends on the method that you feel comfortable with.
Personally, I would find a decorative cup that goes over the circle to hide the raw cut.