From what I understand, lice and mites are not very active in cold weather
Not true, lice and some mites live their whole lives on the bird where it is nice and toasty warm.
I'm so sorry you lost 2 birds, but have to agree that it wasn't just the cold that killed them. The cold
can stress an already sick bird, but heating the coop won't fix the underlying cause of death and in some cases may make it worse.
It can be
very difficult to figure out exactly what ails a bird once they have some disease.
Many birds carry disease but show no symptoms until stressed enough to weaken their immune systems.
A necropsy and pathology testing at a reputable lab is about the only way to know for sure....not possible for most folks. Gross DIY necropsy may offer clues if you're up to it and have a good understanding of what you're looking at....many can't do this either.
Good space and ventilation coupled with a good diet should keep birds healthy enough to with stand the cold. Assess your housing and feeding regime.
Check
all your birds for external parasites:
Google images of lice/mites and their eggs before the inspection so you'll know what you're looking for.
Part the feathers right down to the skin around vent, head/neck and under wings.
Best done
well after dark with a strong flashlight/headlight, easier to 'catch' bird and also to check for the mites that live in structure and only come out at night to feed off roosting birds.
Wipe a white paper towel along the underside of roost to look for red smears(smashed well fed mites).