You only need 2 conductor plus grnd NM-b. 10/3 NM-b is 3 conductor plus ground
As pointed out by exranger06, #10 NM-b is too small
As to what wire to use from the wall to the compressor, #10 THWN is fine in FMC. Or you may be able to find a premade whip at the store.
Also, if the compressor is more than 50' from and not within sight of the breaker panel, you will need a local disconnect.
yes because of inflated numbers. This is why one must check the motor nameplate. The OPs motor however, IS a 5HP rated motor so he must size the wire based on NEC FLC table amps of 28a.

This
actually they do.
heres one rated for 250v and 5HP. you wont like the price though ($200-$300+)
https://www.hubbell.com/hubbell/en/...e/IEC-Devices/HBL330P6W/p/159377?PN=HBL330P6W
https://hubbellcdn.com/literature/Wiring_WLBP005.pdf
you can't compare the 2. Thats a false equivalency. The wires on listed equipment have been tested in a lab, they aren't ran inside the walls of a building next to flammable materials and they often have a higher temperature rating (105° instead of 75° or 60°).
exranger06 already explained the wire sizing requirements for motor circuits. #10/x NM-b is too small for 5HP
As to the breaker sizing, you can use a breaker rated up to 250% of the FLC table amps. But a 30a breaker may very well work and not nuisance trip.
Umm, what?

I've seen numerous 5HP nameplates and they all have an FLA between 20-23a. The NEC FLC table lists 5HP as 28a and thats what the wire must be sized by NOT the nameplate FLA.