Yes of course. Eggs don’t really require refrigeration except that commercially produced eggs are typically washed so the natural coating is removed and it is better to refrigerate them because they already are not usually super fresh when you get them from the grocery.

People who keep chickens and gather eggs daily which are used within a week or two keep those eggs in a basket at room temperature in pantry or on counter.

Definitely

Yes

Yes

Put them in the trunk or the back part of your car

Yes.

They'll be fine today

yes

Basic question. I need some eggs. I'd really like to get them before a meeting today, but no fridge - would have to keep them in my car. As long as they're not in front of a heat source, would they be fine? It's cool out, no sun to warm the car, etc.