Electrical – nything in the NEC that would prevent me from attaching MC cable with ground to EMT with a compression coupling

Before going through all that I'd check:

1. If the existing metal boxes are already grounded (See How should I connect to a short ground wire in a metal box? and maybe Should old knob and tube wiring be replaced? )
2. If the old cables will come out cleanly from below. Fish a string as you yank out the old cable.

Note:

1. The NEC's unsupported rule does not apply inside walls. You get a free pass on that one.
2. You can put an AFCI breaker on the two wire circuits, no problem.
3. You can put a three prong GFCI on the two wire circuits, as long as they are labeled as such.

Is that acceptable for new construction? No. Does anything compel you to fix it now? No.

I'd check very carefully that the insulation on the black & white wires is unaffected (sounds like it is). Then slip something over to mechanically protect those wires. A stiff bit of plastic sheet rolled up and slipped through the joist should do it.

Then caulk the hole for airflow purposes, and consider replacing the entire return plenum: there is no caulk and it leaks air. At best you're getting basement air into the plenum because the pressure is lower.