My main question is, how do you guys handle the extra companions at camp?
I ignore the ones that I'm not actively using, outside of the few times where bringing them along actually matters.
Does it matter or affect the main story if you keep the extra companions in your camp idle and ignore them completely?
No.
Is it better to dismiss all the extras from the camp instead?
It literally doesn't matter.
Apparently some events proc automatically even if you ignore the companions. (Ex. Astarion's first camp event).
Does this happen often for passive camp companions?
Yes. Literally all companions will passively progress through their stories without you needing to do anything. Heck, most of them you don't even need to talk to them at all or even Long Rest (The only ones you NEED to Long Rest for, will be forced upon you when you transition between Act locations - Much like other key camp scenes)
You only have to actively speak to or bring along companions in very few situations. Such as needing to talk to Gale once to make Elminster spawn in Mountain Pass. Karlach needs to be in your party for Dammon to upgrade her heart. Also, Act 3 companion quest finales require their respective companions. (Also, Wyll needs to go into the Mind Flayer Colony if you want to receive the Infernal Rapier, but is not necessary to progress his questline)
Outside of that, you're free to ignore them as much as you want.