You would be able to use just 12/2 bx but I always use MC cable or 12/3 and tape the white conductor green and use that as a grounding conductor, I never like to rely on the BX jacket for the grounding.
You don‚Äôt need a neutral.
The stove probably has a 120v timer on or digital display that‚Äôs why it required the third current carrying conductor.
DO NOT make any adapter for your compressors plug, get the right receptacle preferably a single receptacle and make sure the power is off and be sure your equipment is grounded.
However, it is also better to use an outlet plug that fits your appliance instead of using an adaptor. Adaptors weaken the physical connection and can heat up when used too much.
Your compressor is wired with three-prong, and that is the only appliance you plan to use on the circuit, so I would wire it for three-prong with two hots wires, each carrying 110-120 volts. However, there are appliances with four-prong plugs. You might want to wire the neutral to the box, anyway, cap it off in the box while the only appliance you have is the compressor, and then have the neutral there for future appliances so you can safely change out the outlet plugs when needed. Check your local code, but you might also be able to install two outlet plugs (one three prong, one four-prong) so long as you do not use them at the same time.