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As long as the temperatures never drop below 60 degrees. Anything below this will damage, or kill it.

This article will give you more information on their care: https://www.gardeningknowhow.com/ornamental/flowers/desert-rose/caring-for-desert-rose-plants.htm

Over time they can adapt to the wind, and they are moderately salt tolerant. The important thing will be to keep them from getting cool. They are not cold tolerant at all.

This article will give you more information on the care of these plants: https://www.gardeningknowhow.com/ornamental/flowers/desert-rose/caring-for-desert-rose-plants.htm

It isn't as bad as it seems. Rough times can cause it to go into a semi dormant state. This article will help you with proper care so that it is looking its best in no time: https://www.gardeningknowhow.com/ornamental/flowers/desert-rose/caring-for-desert-rose-plants.htm

If it still has a root attached, then you do not want to cut this off. This will give you a kickstart on the process, instead of waiting for it to regrow a root that has been cut off. So I would not cut the root off at all.

If, in fact, you meant the stem and not the root then you will want to cut this at an angle. This article will give you more information: https://www.gardeningknowhow.com/ornamental/flowers/desert-rose/desert-rose-propagation.htm

You probably could unless the rot is too deep in the plant. In that case, you could always find a healthy part and root a cutting.

It is probably reacting to the repotting, which was probably unnecessary as Desert Rose does just fine very tightly potted so it is not as easy to inadvertently over water. Be sure to water very sparingly now.

Yours would benefit from pruning back the many leggy stems.