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Yes, This can be an option. This is called Pot-in-pot production.

You will implant an equal size container with good drainage in the soil as the one that the tree is in. Then just plant the other container directly into this.

This set up will allow you to move the trees into different locations as needed without disturbing the roots.

I found a website that will help explain things a bit further: http://www.uky.edu/ccd/production/system-resources/nursery/pot-in-pot

Cherry trees are not picky about they type of mulch used. It will be up to you, depending on how you would like it to look. Some plants can be sensitive to type, but this will not be much of a concern here.

Most causes for leaf curl will be treated in Autumn, yes. It will be necessary to pin point the exact cause, though. This could be viral, bacterial, or simply an insect problem. This means that the treatment will vary depending on the exact cause.

This article will get you started: https://www.gardeningknowhow.com/edible/fruits/cherry/cherry-tree-diseases.htm

A picture would be really helpful here. It can be saved easily as long as there is still some attached bark at least in one area. If it is completely ringed, it can be saved, but it will be a little more difficult.

This article will help, in the meantime, until a picture is provided: https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-fix-girdled-trees.htm

Covering tree wounds with latex paint or a tar compound is no longer recommended in most cases. Paper and plastic tree trunk wraps are available at most hardware stores and may prevent deer, rabbits and rodents from chewing the bark in winter. They also help prevent the trunk from splitting due to direct sunlight causing freeze and thaw cycles on the south side. Remove in spring. Try stapling some hardware cloth to stakes to keep the deer away. There are taller, ready-to-use tree protectors available on the web. They are often used in state and national parks where antler-rubbing, browsing animals are a problem. Here is more information: https://web.extension.illinois.edu/state/newsdetail.cfm?NewsID=11716

They would do best in a greenhouse, where they can receive full natural sunlight all day. They may also do much better if you go ahead and plant them in a 1 gallon container. A small cup will not last very long, and you will have to repot again within a couple of weeks. A one gallon container will provide enough room for, at least, the first six months of their life.

These will not do very well indoors without supplemental lighting. They will require about 100 watts per square foot of trees inside.

After this first year, they will be hardy enough to leave outside since they are hardy in your area. You can go ahead and plant in the final container, or in ground after the last frost in spring.