The first one might be an Echinocereus. The lumps are probably buds that aborted and then continued to grow as pups.

The Haworthia would like a bigger pot but don't go mad. You can take off the pups if you want more plants, or leave them if you like it as a clump. Old leaves do die off, but I find they die off quicker if you water too heavily or wait too long between watering. Its not a Ferocactus, don't water it twice a year with a hosepipe ;)

Not sure what you mean by "non-soil". You mean peat? Not ideal for the Haworthia, not good at all for an Echinocereus. I use actual "soil", a sandy loam with added coarse material such as gravel, pumice, etc. Others use "non-soil" like coir, which seems to work well, or bark-based mixes which might not be so good. Check the roots when you repot, remove any which are dead and just keep the thick fleshy white ones.

Oh, I know they're pups, lol, I was talking about the way the cactus seems to be twisted (maybe it doesn't show that well in the picture, but it is). I mean, I don't think the pups are tumors or something. X)

I've never heard that before..Why would the buds abort? Trauma?

Lazy as I am, I don't think I'd wait that much before watering, lol. I mean "non-soil" like gravel and..well. Gravel.

I add perlite to my soil (pre-mixed, Miraclo-Gro, but I don't really like it), but would an addition of gravel or pumice be helpful in making the soil not float when I water?