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Regardless of where you live, the best time to prune is in late winter or early spring. This article will help:
https://www.gardeningknowhow.ca/ornamental/trees/crepe-myrtle/best-crepe-myrtle-pruning-time-when-to-prune-crepe-myrtle.htm

This is sooty mold, a common problem with crepe myrtle trees. Here is an article that you may find helpful: https://www.gardeningknowhow.ca/plant-problems/disease/how-to-get-rid-of-sooty-mold.htm

You should be able to, but without knowing the exact shape of the tree, we would not be able to tell you how this would affect the future shape of the tree. It would be worth it to have a landscaper who is experienced with crepe myrtles come and look to make sure that the tree will retain a pleasing shape if this sort of pruning is done.

Crepe myrtles can easily take this kind of pruning without endangering the tree. We normally advise against this kind of pruning, as it can deform the tree visually, but it sounds like you do not have a choice. Prune it in late winter or early spring and cut out the branches you need to with a sharp, clean shear or saw, depending on the thickness of the branch.

It sounds like you may have some sooty mold left over from the pest problem (these two issues normally go hand in hand). The sooty mold will not hurt the plants, though it does make them look unsightly. In the spring, apply a fungicide to the areas. This should help clear it up and make it look better.

You can prune them while they are dormant. Some people prefer to wait, as pruning while dormant can decrease the blooms in the next season, but it will not harm the plant.