I have quite a few metal junction boxes in the unfinished basement of my 1940's house. Some have new NM-B running into them, some have the original 2-Wire Romex (with braided cloth insulation), and some have both.

When did the NEC require that metal junction boxes be grounded? Some of the metal junction boxes I have are really quite old with (what looks like) soldered and taped splices, and I'd imagine they were done to code. Am I required to bring these junctions up to the current code by putting them in PVC boxes?

I have one set of exposed splices in my basement, with 3 #14 non-grounded Romex wires splicing together with wire nuts. Because I have no ground path back to the fuse panel, I cannot instal in a metal junction box. Therefore, I have to use a PVC box.

Can I use a plastic switch box, or a round ceiling box, or does it not matter, as long as I have the correct capacity for the wires and connectors?

Given you have ungrounded cable, you can probably safely assume there was no necessity to ground the boxes at the time they were installed. As long as you're not upgrading the circuit, you do not have to change this at this time.