When replacing a starter (or reinstalling the old one), should the pinion that powers the flywheel (is that actually the flywheel?) be greased or not?
Basically the answer is "no". You should leave both the ring gear on the flywheel (standard shift tranny) or flexplate (automatic tranny) and the gear on the Bendix dry. As @elmerfud stated, as long as the starter is shimmed to spec for its engagement, you'll have no issues with it. Both the ring gear and the Bendix gear are built to take the punishment.
Also, if you consider the overall use of the starter gears in comparison to the rest of the vehicle, they really don't get used much. Besides, if you put grease on these gears, where it is at will attract dirt and abrasives. This will cause wear if you are not doing a continual maintenance on them (cleaning both gears and regreasing on a regular schedule). This is something not only do you not want to do, it's something which doesn't need to be done.
This is one of those solid "maybe" answers. If you have access, I would consult the manufacturers assembly/repair manuals for your vehicle. That will be the most correct information for you.
In my experience it is generally not done, but in most cases a small amount won't hurt. It is not a continuously lubricated part nor is it usually something that is lubricated on a maintenance schedule. If the starter is shimmed to spec for its engagement with the flywheel you should have no issues with it engaging or disengaging.
Yes, always on a new starter you should get a small brush and put a good layer of moly grease on the pinion. It's not done during assembly simply as a way to save money because it won't cause problems in the short term. This is simple common sense, they're gears like any other that will benefit from lubrication. Nissan products especially seem to eat ring gears, lube from the factory would have saved so many people so much money.