Do you have a picture of the rest of the plant?

Echinopsis (Lobivia) arachnacantha

Euphorbia, not a cactus. I don't know much about Euphorbia, so I leave the rest to those who do.

Gymnocalycium ragonesei for the last one?

For the opuntia, do you think it would be possible for me to take cuttings to replant, or use some of the pups it's dropped? Yes, but it will probably take a long time before it has grown into a plant worth looking at, and it is easy to find another mature plant. But it could be interesting as an e...

The first is probably a severely etioliated Opuntia microdasys. I would not try to save that one.
The last is Mammillaria vetula ssp. gracilis.
The rest are different Mammillaria species, except no. 3, which I don't recognize. Cylindropuntia?

hob wrote:yellow would indicate marsoneri

Especially if it is (as I suspect) a mass-produced plant. R. marsoneri is not uncommon in flower shops.

CactusFanDan wrote:I'm thinking it could be R. violaciflora, though.

The dried up flower looks yellow to me, and violaciflora has pink flowers (more red in the form knuthiana).

2: Isolatocereus dumortieri?

dattdude wrote:Parodia Scopa have white spines right?

Some have only white spines, some have both white and red(dish) spines.

4. Thelocactus hexaedrophorus

Seems correct.

I think it is a hybrid called 'Blaze'. Identifying clones from pictures alone is difficult (there could be other hybrids that look like this and that I am not aware of), I suggest you put a question mark after the name if you decide to call it 'Blaze'. Important if you pass it on to someone else.

The big one is Austrocylindropuntia subulata.
The very spikey one is a Parodia (Notocactus) of some sort.
Photo 4 and 5: probably etioliated Echinopsis chamaecereus (most likely hybrids).

cacti005.JPG, left: Parodia mammulosa(?), right: a Gymnocalycium.