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This appears to be a species of crassula, which is related to the jade plant. These succulents are easy to care for and have th same requirements as most succulents. Here is an article that will help: https://www.gardeningknowhow.com/ornamental/cacti-succulents/scgen/cacti-succulents-inside-your-home.htm

This is, most certainly, a type of crassula. This is the jade family. There are so many cultivars, and many of them are very hard to tell apart so pin-pointing the exact cultivar will prove difficult, but they all have very similar care.

This article will tell you how to care for jade plants in general. This will apply to almost all forms of crassula: https://www.gardeningknowhow.com/houseplants/jade-plant/jade-plant-care.htm

All crassula are considered highly toxic to cats. Please keep these out of reach of the cat.

Yes, actually, that will be the best course of action. I see that most of those succulents are not hardy in your area.

I also see a type of jade that will not be as tolerant of the dry condition as the other succulents. They do thrive in dry climates, but will want just a little more water than other succulents. You will have to balance the conditions to keep them all happy. Still, generally speaking, dry is better than wet in this case.

This article will help you with general succulent care: https://www.gardeningknowhow.com/ornamental/cacti-succulents/scgen/care-for-potted-succulent-plants.htm

This article will help you with jade care:

Some dieback is normal, but this does seem like a lot. If it's not mushy, it may not be rot. You can prune off the dead stems and notice if there is any soft rot. If so, you can unpot the plant and remove the rotting parts and repot the healthy foliage. To be on the safe side, you can also cut off a couple cuttings to air dry for a week, then pot up to make a new plant.
Check your potting mix. Make sure you have a good succulent mix that is quick draining. Also let the soil dry out between waterings.
https://www.gardeningknowhow.com/ornamental/cacti-succulents/jade-necklace/jade-necklace-worm-plant.htm