I believe the factors are alleged to be many, including dry air in heated rooms which irritates nasal passages, many people often huddled indoors which contributes to infection spread, and so on.

Cold temperatures are, of course, involved in all of those, but very tangentially. (Unless one persistently lives in poor heating conditions, which combined with cold outside would likely take a toll on one’s immune system—from my, non-doctor’s, understanding.)

It could do, but there are other known effects at work (such as temperature and humidity) that are known to contribute, and that would also be correlated with vitamin D levels. You’d need to do a study to tease out the causality and the relative contribution of all those factors (if one hasn’t been done already).

Edited to add: this looks like a reasonable review of the literature: https://www.annualreviews.org/doi/pdf/10.1146/annurev-virolo...

This is a possibility. Or at least it is a plausible mechanism of action for higher susceptibility. If true we should expect seasonal variance in prevalence to decrease as you move towards the equator.