For the armor perks that give you extra armor if you only wear Light or Heavy, do you lose this bonus without a helmet? Say, wearing a circlet instead?

Yes. For the perks to take effect you need the boots, armor, gauntlets, and helmet

Yes, but if you like cloth hoods with armor fx, there is a mod that makes hoods into armor so you get the perk.

I think the item has to be listed as light or heavy armor for it to count with those perks. You can see what type it is in the top left corner when viewing it in the menu.