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Yes, you can split the root ball. The most common way to reproduce this plant is by new growth cuttings, but it will certainly be ok to split it. Here is an article that will help: https://www.gardeningknowhow.com/ornamental/vines/mandevilla/mandevilla-propagation.htm

The only thing I would be concerned about is the dry air. This could cause some leaves to dry out, and fall off. If you can keep it directly out of the air stream then this would be best. If this is not possible, then I don't think it would kill it. It wouldn't like it as well though. Here is an article for more information on their care, and how to tell if it is a dipladania or a mandevilla: https://www.gardeningknowhow.com/ornamental/vines/mandevilla/growing-a-dipladenia.htm

You can prune them heavily. There will be no issue with this. It is best to do this after the flowers fade, and before new vegetative growth starts.

This article will give you more information: https://www.gardeningknowhow.com/ornamental/vines/mandevilla/prune-mandevilla-vines.htm

There are a number of things that you can use for this. I prefer organic in almost anything I do, but there are chemical methods as well.

I like oils such as: Rosemary, lavender, garlic, clove, and neem... All make wonderful, and safe pest controls that you can mix, spray or water into the plant without hurting it.

Here is a link to a collection of articles that will help you find other methods for controlling insects: https://www.gardeningknowhow.com/special/organic/

I am not sure why this would happen with your Mandevilla, but some flowers change colors when environmental conditions change. Perhaps the change in light level made the difference.

I would only cut it back about half way. They can recover, but this is very stressful on the plant. You might invest in some neem oil to prevent this in the future as well.