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It does look like most or all of the leaves were cold-damaged, but the trunk and roots may have survived. First, wait a few weeks to see if any of the leaves perk up. Cut down on watering, because damaged plants use less water and you don't want to give it root rot, and avoid fertilizing it. Then, you can cut off any dead leaves and keep the leaves that are recovering.

If all the leaves seem to be dead and are turning brown or mushy after a few weeks, you'll probably need to prune it back by cutting off the top. Try to cut back until you find a healthy (non-mushy or rotten looking) portion of the trunk. Ponytail palms can often regrow after the tops are cut off, but not always. You'll just have to wait and see.

Try a soil moisture meter. Often available at local garden centers or online for about $10.
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-moisture-in-plants.htm

If the glued-together pebbles are forming a solid soil covering and preventing water infiltration, you may need to remove that covering or replace with an organic mulch.

These are hard to make root when done from cutting. It can be done, but they typically fail. Sometimes they will shoot offsets below the soil line, and these are suitable for using as a cutting, since they have already developed roots under the soil.

This article will help: https://www.gardeningknowhow.com/houseplants/ponytail-palm/propagating-ponytail-palm-pups.htm

This is almost always caused by a past fungal infection. If the plant is not showing any other signs of stress then there is not much to be done. Anything that you do attempt may actually harm the palm. They can tolerate being root bound, and actually prefer it. I believe that it will live longer if you leave it in the container that it is in. as removing will only increase the chance that it spreads faster.

As far as that bandage... If you can manage something that will let air flow through then it may prolong the spread, but I would recommend leaving it exposed, and avoiding getting water on that spot.

Do what you can to save it. There are many ways to stop rot. It would be accurate to say that it was over watered... This is common, luckily there are many methods to removing infections such as this. Peroxide, rosemary oil, garlic oil, thyme oil, copper solutions, and wettable sulfur are all wonderful methods for controlling fungal issues.

It would help to replant it in a slightly bigger pot, and in a tropical mix. This should help your palm make a full recovery. Here is an article that will help: https://www.gardeningknowhow.com/houseplants/ponytail-palm/ponytail-palm-care.htm