Looks to be a Pachyveria or a Pachyphytum possibly hookeri. As for the leaf I'm not sure how to save it and would just gently pull it off so it doesn't rot. Speaking of rot, the plant looks great at the moment but in a glass jar with a soil containing so much peat it may not stay that way for long. It'd be best to switch it to a fast draining soil mixture without much organic material. Search for "Al's Gritty Mix" for examples of what I mean. Should also switch it to something with drainage holes and preferably a type of pot that breathes well, like terra cotta.

It appears to be an Echeveria, but I'm not sure what kind. It is very etiolated (stretched) from trying to reach light in Walmart. As you already know, it has suffered from soggy soil. That's what caused most of the leaf drop. It likely lost much of the healthy root ball to rot also. It looks like it was there a long time, but you have rescued it, and I think you can get it back on its feet. Here's what if suggest: Pot size & type: 3"-4" plastic pot. Maybe the pot it came in. It'll be small enough, and it'll have drain holes. A larger pot will contain too much soil, and too much soil means it'll stay wet too long, and desert plants don't handle wet soil well. Planting medium: If you're going to add your own perlite, Cactus and Succulent mix isn't neccessary. It's usually the same as regular potting soil with a little extra perlite in it- not enough to matter. Mix it with an equal amount of perlite, which you can find right next to it on the shelf at Home Depot. Water: Water only when the top 1/3 to 1/2 of the soil is dry. You have to stick your finger in it every time because it won't always dry at the same rate depending on humidity, sunshine, sir movement, and whether it's actively growing or it's dormant. Sun: It desperately needs sun, but it's been shaded so long that it can burn if it's out too long. Start with about an hour or two of direct sun daily, and over the next few weeks increase it by an hour or two weekly. In the meantime, get it into a window with as much bright, indirect light as possible. Don't worry about the windowsill being cold just yet. Desert plants handle cold better than tropical plants. When the glass gets very cold, the leaves that touch it will get dead spots, so be careful. If you can set it outside, it'll be okay until the first frost. Temperature: Sunshine and correct moisture are much more important to Echeveria than temperature. Newbies often think desert plants must be warm all the time, but they don't. Deserts are dry and sunny, and often quite cold. Fertilizer: Never (soil usually has enough in it already for succulents), or twice a year with 1/4 strength, but if you repot it often enough (to keep the pot barely big enough for it) it'll get what it needs from the new soil each time. Right now- it will need some water. Be conservative since you have a poor mix and no drain hole, but you must water it. If you don't, the existing roots will die back and it'll have no way to absorb water when you do water it, and then it will rot. I'd recommend a spray bottle set to "stream" and squirt some streams of water right down into the mix near the roots. Hopefully that mix isn't too hydrophobic to absorb and disperse the little water you give it. This plant will do most of its growing in the fall and spring. It'll be dormant in tje hottest part of the summer and coldest part of the winter. It'll need more water when it's growing, and it will be less tolerant of too much water when it's dormant. If it isn't growing, don't be fooled into thinking water is the key.

I would not remove any leaves from the clipped rosettes. Remember that the leaves are stored energy (water and food) for the plant. The same goes for the stem left behind. You can always trim that last rosette off later, hopefully after you get some back-budding and more rosettes. I like a longer stem, an inch or so, to stabilize the cutting, but that is just my preference. Sometimes that is not available. Let the cutting dry for several hours or over night. Then insert the stem into DRY potting mix, or lay the rosette atop the mix. Your plant takes up water through the roots. If there are no roots, no water is given. Otherwise you may get rot. Here is a rosette that I just rooted. (just rained, so that's why it is wet) It's ready for me to plant in the yard. See how it is sitting on the mix? And here is the plant I took it from. It has sprouted a pup that I will root eventually. I will probably use those leaves for propagation, too. But I am in no hurry so I will see what happens with this plant. (The stem is black because I put cinnamon on it after cutting.) Leaves will sprout just about anywhere. I'm not kidding. This one fell off a plant and landed between pots. I found it when I was trying to clean up some. LOL Often, my leaves sprout roots while laying on the counter. That's because I get busy and forget about them. Other times I lay them on my potting mix. Sometimes I tuck the end into the substrate, other times I do not. Some leaves simply do not grow, so don't fret if some of them don't. Some people mist them, but that has lead to rot for me. I only mist if there are roots. Again, no roots, no water. =) After I carefully arranged leaves in this pot, the wind mixed it all up! Leaves were buried and I probably didn't dig up all of them. A lot of the mix blew out, too. But they are doing their thing! I hope this helps some. Gardening should be fun, so don't let it stress you out. Some plants are easy for me, and others I cannot grow for the life of me. Trial and error - with good advice from friends here - has helped me choose. Good luck! You have a very nice looking plant.

Succulents needs are quite similar. If you assure fast draining mix, use a container with drainage holes and expose them to good light - after acclimatizing first - you don't even have to know exact names. Saying that, some will grow quite well in average light - for example Sansevieria, Haworthia - but low light will make them grow very etiolated. Maybe you mentioned in some of your other post your general location or at least growing zone, but it would help to add to your name - see what others do...