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This is best transplanted when the shrub has gone dormant for the season. Here is an article with more information about the shrub: https://www.gardeningknowhow.com/ornamental/trees/crepe-myrtle/crepe-myrtle-care.htm

This is because you have a tree that is called "Tamarillo". This is not a crape myrtle. It was just a case of mis-identification. This tree is in the Solanaceae family.

Here is some more information on the tree: https://www.gardeningknowhow.com/edible/vegetables/tree-tomato/grow-tree-tomato-tamarillo.htm

This can start to happen naturally during cold weather. If this is not the case then I would suspect a fungal infection. If you do suspect a fungal infection, then prime the soil with wettable sulfur and dolomitic lime. This article will give you more information on the care of the shrub: https://www.gardeningknowhow.com/ornamental/trees/crepe-myrtle/crepe-myrtle-care.htm

Late winter, or early spring is the general consensus. I would prune during these times. This article will help: https://www.gardeningknowhow.com/ornamental/trees/crepe-myrtle/best-crepe-myrtle-pruning-time-when-to-prune-crepe-myrtle.htm

The moss will not harm the tree, and yes, this is perfectly normal. They will regrow after dormancy. This article will help: https://www.gardeningknowhow.com/ornamental/trees/crepe-myrtle/crepe-myrtle-care.htm

Ideally, the best time to prune these is in early spring or later winter, during dormancy. This article will give you more information: https://www.gardeningknowhow.com/ornamental/trees/crepe-myrtle/best-crepe-myrtle-pruning-time-when-to-prune-crepe-myrtle.htm