I'd honestly go 16" apart(I'm assuming this is for running the furring vertically? Minimum 8 feet?). 5/8 drywall isn't light and humidity can affect the wood and cause it to warp, leaving you with more problems.

The better it's fastened the better your finished product is gonna be. It seems like a lot but do it right now so you don't have to do it again later.

As for tapcon size, 3/16 is fine. For length id go with a 2 1/4" tapcon(also depends on your location as a 2" will do but for my region they don't have them). Also, get a flat head Philips or star drive, no hex heads as the hex heads will either stick out or youll have to drive them so deep that you can split the wood