Your car sounds like an excellent candidate for boost. And there's plenty of experienced hands around here to help you out, poke fun at you from time to time.
The second O2 sensor is basically there to monitor catalytic efficiency. It's expecting to see a "lean" condition most of the time, indicating that the converter is doing its job.
That being said, a "high-flow" converter will probably be sufficient for your needs. I recently passed CA emissions using a 2.5" Magnaflow on my turbo'd 1.6. And while removing the converter might theoretically gain you an extra 2 HP at the redline, modern honeycomb converters really don't deserve the bad rap they've inherited from their predecessors. If you're set on removing it, you may be able to fool the ECU by leaving the sensor in place but diking out the sense line and feeding it a fixed reference voltage instead. Easy enough to do with a couple of resistors. Actually, I really ought to design and sell that...
You'll still want an O2 clamp for the primary sensor if you decide to keep the stock ECU. Just my opinion, but it's good practice for *any* car where you're trying to exert control over the stock fuel system.
Everything GreddyMX5 said is pretty much bang-on. No Miata from '90 to '05 has ever had a MAP sensor, except possibly the factory turbocharged Mazdaspeed models. Never worked on one of those.
Those coolant lines are your heater hoses. While you're replacing them, it's not hard to slip some Summit Racing brand sleeve around them. You are replacing them, right?
You'll also want to wrap pretty much the whole brake system in a heat blanket. Somewhere around here there's a picture of somebody's molten master cylinder cup. Very bad...
Welcome to the club.