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It is difficult to identify specific Succulents since there are so many. You may be able to search identification sites and find the type.

Overwatering is the most common reason for loss of a succulent. The soil needs to be light and well draining. Too much moisture and the plant will rot.
Find a nice sunny window for your plant.
Here are some links with more information.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/scgen/succulent-plant-info.htm
https://www.gardeningknowhow.com/ornamental/cacti-succulents/scgen/exotic-showstoppers-beautiful-succulents.htm
https://www.gardeningknowhow.com/ornamental/cacti-succulents/scgen/cacti-succulents-inside-your-home.htm

Your plants look quite healthy! The biggest key to success with Echiveria is well draining soil!

You can actually remove the flower stem at the base of the plant if you feel it is doing damage to the plant.

Echiveria will generally bloom once a year.

Here is a link with more care information.
https://www.gardeningknowhow.com/ornamental/cacti-succulents/echeveria/caring-for-echeveria-plants.htm

Yes, it will likely recover on its own by sending up new shoots. Just give it time. Care for it as usual until you see new growth. Then you can remove anything that is dead. These plants are well adapted to survive, and in the wild are constantly stepped on. They usually recover with no problem.

That is a flower cluster! Let it bloom. This only happens if it has the perfect environment to allow it to do so. Enjoy the blooms!

I see the picture now. This succulent will be best in a larger pot with succulent, or cactus mix soil. Both the new top, and the existing plant will recover, and regrow. Plant the top, and care for it as you would the original, and it will become its very own plant.

It is best to transplant these into a succulent mix. it will likely root itself with normal care, as you would the mother plant.