I see that T. schmeididickianus has a white flower, but the structure of the spines is very different than those on the plant I've got. I think I'll send a picture to the nursery and see what they think. Thanks for the Miles and Miles link.....not that I need another source of temptation!

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.

It looks like the San Pedro Cactus (Trichocereus pachanoi). It's kind of scarred and beat up, but I don't think it's necessarily sick. It appears that it may have scale insects, which may require a systemic insecticide. The white powdery stuff is mineral residue that builds up from watering the plant. cd

Likely Euphorbia resnifera. You can remove a column, let it callus over a bit and plant it in fairly dry, well draining soil and it will often reroot (if not cold and damp).