1. Can MC cable be run down a cement wall to a receptacle in a residential basement?
2. Can you connect a furnace circuit to a furnace in a residential home by dropping down with MC cable from a box in the ceiling where it is spliced to NM cable?
3. If the answer is no, what are the code reference?
4. Subject to physical damage? If so, how do you protect it?

This is the old "physical damage" question and most AHJs have their own opinion of what it really is.
Some places will allow NM on running boards down to some arbitrary height above FF. Others want EMT. Most are somewhere in between (Smurf, MC/AC etc)
It really comes back to how exposed it is to physical damage and is that damage "severe"?
I am one of those middle guys. A metallic cable, Greenfield or smurf would work for most of these cases. I never thought much of Romex on running boards, particularly much below 6'6 but that was the law of the land in Maryland when I was there, down to about 48" or even 36".

George:
By your numbers:

1: Unless physical damage is evident, I see no reason that it is not compliant. No, I have not seen anyone doing this around here. Professionals use a raceway. (EMT/PVC/etc)

2: Yes. As long as the MC is supported, and not subject to physical damage.

4: Protection by a raceway (EMT/PVC, etc) or any other suitable (to the AHJ) method I would guess.

George
I'd say use PVC sch 80 for protection when subject to physical harm. (no bonding worries).
Example : underground coming up a outside pole.
Yoopersup