Many many many hours of effort (maybe fabrication work?) and troubleshooting to get everything harmoniously working, Florida is super lax on motoring laws so that shouldn't be an issue.
After seeing project cars do unexpected things and owners scrambling for hours and missing runs and sessions, it's a labor of love with small reward. My money is on reliability, keep it simple stupid and maximize seat time, keep it stock.
While a Civic Si or TSX motor would be awesome, (~200hp @ 7k rpm) and reliable, the potential issues won't be the block itself but rather all your wiring, sensors, everything mating up well mechanically (no rubs or unforeseen stresses), and ECM. Custom dashboard or a bunch of stick on gauges? Custom exhaust setup? This is not a summer project unless it's your full time job and you have access to a good shop, most people spend years on these kinds of setups before everything is dialed in.
I'd spend the $8k on setting up the Miata real nice, good chassis to start spending up to $5k, nice set of revalved koni/bilsteins with quality springs from any of the half dozen companies that offer them off the shelf, go through all the bushings, roll bar, quality tires and brakes, full fluid flush. Assuming this is a casual track/autox car I'd say maintain DD-ability so you can drive it to the track without stressing yourself out and avoid the costs of a tow vehicle because you've gone full stipped and caged.
For me the name of the game is seat time, you want zero excuses to not go out and flog the car.
No, S2k uses an F20C
Wiki is a fantastic resource for stuff like this, the K Series is primarily used for higher trim Integra/RSX/Civic, and Accord/TSX/ILX with a few larger cars/cuv using it as a base engine from the early 00's onwards. Good for 160-220 hp depending on what model which should be a solid upgrade for an NA/NB
https://en.wikipedia.org/wiki/Honda_K_engine