How do we determine if a hydrocarbon is planar or not?

Answer 1

Experimentally. 1, 3, 5, 7-cyclooctatetraene is not planar, and we know this from experiment. It's also defnitely not aromatic because it has a number of π electrons which is divisible by 4 (Hund's rule), and planarity goes hand-in-hand with aromaticity with the exception of cyclopropene. On exams, you'll be told whether it's planar or not, so don't worry about it.

There are computational methods of figuring this out, but that's more the remit of working physicists.