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Yes, I would say Jasmine Vine.

https://www.gardeningknowhow.com/ornamental/flowers/jasmine/growing-jasmine-plants.htm

Yes, it is possible to grow jasmines and peppermint in containers. Here is more information:

https://www.gardeningknowhow.com/ornamental/flowers/jasmine/repotting-jasmine-plants.htm
https://www.gardeningknowhow.com/ornamental/flowers/jasmine/growing-jasmine-plants.htm
https://www.gardeningknowhow.com/edible/herbs/mint/growing-peppermint-plant.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/using-topsoil-and-potting-soil.htm

Gardening Know How doesn't sell plants, but you may be able to find both plants you are looking for in gardening stores near you, or through online stores. Many gardeners also get plants from other gardeners in their area- many are willing to share cuttings and seeds with other gardeners.If you can find a local gardening organization, they could probably help you find places to buy them or other gardeners who have these plants.

This article should answer your question:
https://www.gardeningknowhow.com/ornamental/flowers/jasmine/growing-jasmine-indoors.htm

You can cut them back By at least half if you wish. This will only help promote better growth the next season. Here is an article with more information: https://www.gardeningknowhow.com/ornamental/flowers/jasmine/trimming-jasmine-plants.htm

More than likely these are related issues. Spider mite damage can appear as white speckles but if it is a true white then I would consider a powdery mildew issue. Try reducing the humidity around the area, if possible. Otherwise, I have a recipe that I use to both regenerate plant soil, and to kill and protect against insects, and fungal pests. My recipe is as follows:

First, you will want to prime the soil with dolomitic lime, then spread wettable sulfur to Kill off any infection in the soil. I Also like to spray the plant itself. Mix a few drops of one or more of these ingredients into warm water with a spoonful of coconut oil: Rosemary oil, lavender oil, neem oil, garlic oil, cinnamon extract/oil and for really bad cases, also add clove oil.

Here is an article for other ways to kill pests: https://www.gardeningknowhow.com/special/organic/natural-home-pesticides-organic-garden-pest-control.htm

I think the plant you're referring to is Carolina Jessamine, Gelsemium sempervirens (some people do call it Carolina Jasmine, but it's not a true jasmine). This plant is winter-hardy in zone 7, so it should do fine. It is not recommended to cover them for the winter, because the leaves are semi-evergreen and will still be photosyntheiszing in the winter. Placing a wind screen near them is a possibility. They will also be protected if they are near trees or shrubs that break the wind.

This plant can be pruned if desired, but the best time for pruning is just after they've finished flowering. They won't benefit from pruning in the fall.