How did the Numenoreans survive in the Cave of the Forgotten?
 

When Ar Pharazon sails to Aman to claim immortality, the Valar laid down their government of Arda, until Illuvatar intervened and made a huge wave to drown Numenor. Illuvatar also made the hills of Calacirya collapse on the Numenoreans. Akallabeth never explicitly state that the Numenoreans died when the hills collapsed on them, only that they were imprisoned. They would be imprisoned until the Battle of Dagor Dagorath, when Morgoth breaks free from confinement and resurrects his armies. I think this implies that the Numenoreans must have some way or another survive under the caves of the Forgotten. How did they do it?

I don't see why the Valar could not in that circumstance have indefinitely halted the aging and metabolic processes for the Númenóreans. Morgoth did it to Húrin when the latter was prisoner in Angband, basically keeping him in what looks like a state of mental awareness with physical stasis.

The other option is that their bodies did die, but their spirits were prevented from leaving the cave and going on to whatever awaited Men after, until they'd been specially judged.

Of the two, I think the former is more likely.

Well, Hurin's case supports the former, but the Paths of the Dead support the latter. The cursed people died - or their bodies did, but their spirits stayed.

If the Valar could make the Numenoreans live in a cave forever it seems they did lie to them when they said the Numenoreans couldn't become immortals and so the Numenoreans had a legitimate grievance.

If in fact the Valar did indefinitely arrest the physical processes of the Númenóreans in that case, it would have been with the consent of Ilúvatar, who himself had intervened in that situation. At any rate, a suspension of bodily function is far different than true immortality.
Alternately, it could have been the power of the One alone that was responsible for the fate of the mortals there.