Yes, the APS was restartable, as in fact most pressure-fed hypergolic motors are.
Right. I can't think of a pressure-fed hypergolic rocket that
isn't restartable, and can be fired an arbitrary number of times until its propellants are depleted. The only complication would be ensuring ullage, i.e., getting the propellants in partly filled tanks to the bottom where they can be piped off. This is usually not a problem on the first burn when the tanks are full (or in the case of the LM ascent engine, experiencing gravity) but restarting any kind of liquid-fueled rocket requires either the propellants to be enclosed in positive-expulsion bladders or an RCS "ullage burn" to push them to the bottoms of their tanks.
The difference between the two LM stages is that the descent engine could be throttled while the ascent engine could not be. The CSM's SPS engine was also fixed thrust.
I don't know why I forgot the space shuttle main engines. Yes, they were throttled to decrease acceleration during Max-Q, the period of maximum aerodynamic pressure. The solid rocket boosters also "throttled down" during this time, though it was not commanded in real time but built into the way the propellants were cast into them.
Closer to the original question, the space shuttle main engines also throttled down just before cutoff to limit acceleration to 3g. So yes, there
is at least one case in which engines are throttled back to limit acceleration as the vehicle loses mass.