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I believe this is a Tradescantia sillamontana or a close relative. It is in the spiderwort family:
https://www.gardeningknowhow.com/ornamental/flowers/spiderwort/growing-spiderworts.htm

You can tell that the flowers look similar to those shown in the article, but the leaves are quite different. That's because unlike many in that family, the plant you have is a succulent as you've guesses. Therefore it needs less water and is very drought-tolerant.

Although the original might not re-grow, it will, certainly, shoot up new pups to replace it. Often, they will produce several more than just replacing the original.

I am noticing that you soil seems to be a heavy clay soil. This will restrict the growth of these succulents. You can use a potting soil, but they like cactus, or succulent mixes best. They need quite a bit of aeration. Try placing these in a window that receives bright light all day, if possible.

There are many succulent mixes available. You may repot, but have care with the roots. it is best to make sure the soil is very wet when you do transplant. This will make it easier to get all of the soil off without harming the plant. After that, you can treat them as you normally would.

This article will help: https://www.gardeningknowhow.com/ornamental/cacti-succulents/scgen/cacti-succulents-inside-your-home.htm

This would depend on the exact type of succulent, but generally you will want it to flower first. Knowing whether it was male, female, or both will be the next step. After flowering you will want to pollinate, or if it is self fertile, then this will skip a step. After, you will wait until the flower drops, and the seed pod swells. Once the seeds mature, you can then harvest them for planting.

Try doing an internet search for succulent nurseries. I saw numerous hits.