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It will be very hard to tell, exactly, what that is. When it is that young it will tend to look very similar to many others.

In order for them to root, you will need to find out which way is the bottom of the cactus. This will usually be what looks like a "cut" end. The top will be an endless set of spine rings getting smaller and smaller towards the center.

You will need to expose that bottom end to sand, or something that it can have direct contact with, until it roots. Then you can plant it as you would any other cactus. As it is, the spines will prevent the bottom from touching the potting media, so you will have to push it into something soft, and sand would do this.

Cactus pups can be removed and propagated. General pruning is not recommended and can lead to disease or plant loss.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/fairy-castle/fairy-castle-cactus.htm

Disocactus should be grown in a rich but still well draining soil. They should be watered regularly and fertilized as well. This is essential for good growth and flowering. Keep relatively dry in winter or when night temperatures remain below 10° C. Disocactus dislike extreme heat and cold, so they are best green-housed during the depth of winter and heat of summer in hot dry climates such as mine. They also prefer shade to sun, and like a little extra humidity in the air.

This appears to be a grafted cactus with succulents on top.
I would not have a name for this.

This looks like it is a case of phythium rot. This article has more information.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/barrel-cactus/pythium-rot-in-barrel-cactus.htm

What will likely happen is that it will shoot more from completely different areas, but that is not to say that they won't grow them back in that exact spot. They will definitely be replaced, but it will be hard to tell where they will grow next.