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Liriope is quite a tough plant, so a combination of strategies will be most effective. Digging is really the most effective strategy, but is a lot of work, since you have to get as many of the underground roots and runners as possible. Any left behind will attempt to re-grow. You could also try to smother it with weed fabric in the rock garden.

Unfortunately, herbicides aren't all that effective on their own, but those that contain glyphosate or diquat (if available in your area) can help when they're used to kill plants that re-grow after you've dug up most of them. Edging between the rock garden and lawn could help prevent the monkey grass from spreading again once you've removed some of it.

It sounds like they are experiencing a fungal infection. I would apply a little wettable sulfur, and dolomitic lime to the soil. This will help take care of the issue. You can still cut them back now. This will help with recovery as well.

This depends on the variety that you have planted. There are some Liriope grasses that are hardy down to zone 5, but many are not.
I would cover to be safe but always remove the coverings after temps have warmed to above freezing as to not burn the plants.

Monkey grass covers several different grasses including mondo grass, dwarf mondo grass, spreading liriope and clumping liriope.

According to this extension fact sheet, mondo grass is seldom damaged by deer and liriope is occasionally damaged. Bear in mind, although a plant is labeled deer resistant, they will eat about anything if hungry enough, so no plant is totally immune.

https://www.ces.ncsu.edu/files/library/71/Deer%20Resistant%20Plants.pdf

https://www.gardeningknowhow.com/ornamental/groundcover/monkey-grass/monkey-grass-a-spectacular-groundcover-for-the-lawn.htm

You should have no problem growing monkey grass in your area, and it should do fine in the shade.

Here's an article that may help:

https://www.gardeningknowhow.com/ornamental/groundcover/monkey-grass/monkey-grass-a-spectacular-groundcover-for-the-lawn.htm

I would look towards slug or snail damage. Beer traps work, as well as a few other methods. This article will help:

https://www.gardeningknowhow.com/plant-problems/pests/insects/kill-garden-slugs.htm

https://www.gardeningknowhow.com/plant-problems/insects/simple-slug-trap

https://www.gardeningknowhow.com/plant-problems/pests/insects/killing-slugs-with-beer.htm

https://www.gardeningknowhow.com/plant-problems/pests/insects/slug-snail-eggs.htm