It’s generally a good assumption that if a molecule is cyclic, has 6 pi electrons, and is conjugated all around the ring, that it will arrange itself to be flat.
Some antiaromatic compounds with 4- and 5- membered rings can’t really distort themselves out of flatness (too much ring strain!) , but larger ones (like cyclooctatetraene) can, and for that reason, cyclooctatetraene is considered to be “non aromatic” instead of antiaromatic.

That would be an interesting species to try to make. 6-electron nitrogen species are extremely unstable, but that one might have a shot. Do you know if anyone has tried making it?

These values can be deduced based on the old Paquette kinetics measurements for inversion and bond-shifting, or more directly from the photoelectron spectroscopy.

Well, one way would be to make a model. Models are fairly reliable for determining such things. If you make a model of cyclooctatetraene you will see that it’s pretty floppy and can form a tub shape quite easily.
As for other examples… well, here’s one: “The Great Escape” From Antiaromaticity… https://pubs.acs.org/doi/abs/10.1021/ja0291991