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The cactus you have is a very interesting grafted specimen. The pink section on the top is a moon cactus, scentific name Gymnocalycium mihanovichii. It is a mutant variety of that species which cannot produce green chlorophyll and appears pink because of other biochemical compounds it contains. Without chlorophyll, it cannot photosynthesize, so it would die without being grafted to the green cactus on the bottom, which is most likely a Hylocereus cactus (I am not sure of the species).

Your succulent is probably some variety of Echeveria.

You an remove and propagate the new offsets. This can help you extend the life of your cactus.
https://www.gardeningknowhow.com/ornamental/cacti-succulents/moon-cactus/care-of-moon-cactus.htm

At this point, the bottom half may not recover. It has been uprooted on top of what appears to be a fungal infection. The top of the cactus is a separate type of cactus grafted on top. This has a chance for a little longer if you pull them off and sow them in their own pot or pots. They will not live very long, though as they don't produce their own chlorophyll. That is why they are grafted on top of the green cactus.

Infections like this are commonly caused by over-watering. These are very drought tolerant, and like the soil to remain on the dry side.

Here is an article for more information on their care: https://www.gardeningknowhow.com/houseplants/devils-backbone/devils-backbone-plant.htm

It sounds like it started suffering from a lack of light, coupled with a bacterial infection. This can happen when it remains too moist for too long, or if the infection was already there when you got it. Sometimes it can take a little while to show. This cactus appears to be too far gone to save. On the next go, I would immediately transplant into a larger pot with cactus mix. This will allow for proper drainage of the soil.

Here is an article that will give you some more information: https://www.gardeningknowhow.com/ornamental/cacti-succulents/moon-cactus/care-of-moon-cactus.htm

This is very common. Where ever you got it from likely kept it over watered. You can bury up to a little past that, and it will re root from this area. Just make sure to water only when it is almost completely dry to avoid further rot. This article will give you more information on their care: https://www.gardeningknowhow.com/ornamental/cacti-succulents/moon-cactus/care-of-moon-cactus.htm

They can graft to a few other species, but it will be best to be grafted to the cactus that it is originally grafted to.

You can try on many species. Some will take and some will die. Even on a known grafting species, many of them will not take to the graft. This process will be trial and error, so as long as you have several, then you can try to graft to many things.

I would stick to the dragon fruit, or even some species of euphorbia.