Ok first you should take the cactus out? I'm not sure if you have done that yet. That pot doesn't look like the best for cactus:(. You can weave towels cardboard whatever you have around the house to support it when it's laying on its side. You will probably need help. Once the cactus is out you can see what's really goin on. I'm gonna guess all the babies are an attempt of the cactus saving itself because they're all the same size. Sometimes I've noticed not the happiest cactus will do that.

Now that your plant is out time to assess what's going on. Hopefully the roots are nice and healthy. If there is rot you need to be a surgeon and cut that mush off. The columns that have rot you can probably still save. Just cut up the column till you find good flesh. Then let callous for a couple weeks or until there is a nice scab on the end. Keep parts that are callousing in a shady dry spot.

For this cactus I would get a terra cotta pot only an inch bigger than the one you have. Maybe even the same size considering you may have to cut rotten areas off. Succulents and cactus do not like to swim in their pots. They prefer to be in close quarters. Try to find a peatless cactus mix and if you can get some pumice if not than perlite. Start of mixing a ratio of half cactus mix and then half pumice or perlite. If you have a heavy hand with water than do more pumice/perlite.

When you repot plant a little higher in the pot your cactus will thank you. When succulents sit to deep in a pot they become unhappy;). Once everything is in the pot, i'm talking about parts still attached to roots, don't water for at least a couple week. If it were mine I would wait till spring. The cuttings once calloused over plant in a separate pot or the same, but just don't water till roots form. No need to check them every day roots won't pop out for awhile and I mean a month+ don't be shocked if the cuttings take 2 months for roots to form. You will know they will startl ooking better.

goodluck:). If I think of anything I will write again I'm sure there is something I let out. Oh ya sun keep away from the scorching sun while your plant is healing. Any questions feel free to ask people on here are very knowledgeable :).

What are the white 'dots' at the base of your cactus in pic #4? Did you check it?

Stem right behind seems to have some rotted area (the black & soft-yellow looking parts)..