[EndlessAzure]

If you go to a big name shop (like America's Tire/Discount Tire), they won't let you drive off on wheels without TPMS sensors working (a.k.a. sensors on wheels and no TPMS light). If your wheels are loose and you tell them off-road use only, they will probably put tires on and let you go. It's a mandatory safety feature on newer vehicles. Mom&pop/privately owned small shops don't really care.

TPMS on the FRS/BRZ is direct pressure monitoring. It knows directly the PSI in your tires. It switches on if any of the tires is under 25 psi, or if it can't communicate with any sensor (no sensor signal/missing sensor/sensor out of range, or sensor out of battery). I'm not sure what the refresh rate is.

Yes, you can pull the TPMS off of your wheels and put them on any new wheels that are compatible with TPMS sensors.

You do not have to reprogram any sensors unless you replace any one that is already paired with the car.

There are tools that allow you to pair new sensors to the car (See ATEQ Quickset or CARISTA App). Otherwise, you have to do it at a tire shop or dealer.

It's technically not kosher to disable the TPMS system, though it is possible that it can be done with a "tune" using off-the-shelf tools (I'm not familiar with this route).