Could you please post photos directly? You could edit your original post for little while. And number them for easier referencing. Thank you.

Thank you. Now the pictures can also be seen.

Except No. 3 I recently bought this cacti.

Thanks for the reply.
Number 1 looks somewhat like Eriosyce crispa?

Does not look like it so much to me based on the pics I can found on the internet :(

No. 1 I initially thought it would be a Gymnocalycium but the remains of dry flowers make me think that a type of Eriosyce would be more likely. The vendor told me that the cactus had yellow flowers.

Perhaps Echinocactus platyacanthus for #4

#1 looks like a Gymnocalycium to me.

Not an expert at this but I don't see the Echinocactus platyacanthus in #4. It does have a unusual shape more likely because still young which make it difficult to ID (like #5). The inward curving of the needles reminds me of some thelocactus or even ferocactus. Time will tell (or some one else!)

I don't see the Gymnocalycium in #1 as well. Needles are very straight with one in the center. Also it looks tallish. This being said I don't have other suggestion so I may be off as well...

#1 has obvious 'chins' between the areoles which are a characteristic of Gymnos.

#4 also possibly Ferocactus emoryi but it is hard to tell when the plant is so small as ribs can develop later to show the more distinctive shape of the adult form.

#1 not sure what it is, but do you have pics of the flowers? Pretty sure not a Gymnocalycium because I've been under the impression Gymno buds and seed pods are bald.

#2 Mammillaria prolifera

#3 I get the vibe is another Mammillaria. I feel like I've seen this cactus before, but I can't put my finger on it. Reminds me of Parodia scopa. Flowers?

#4 I agree is too young to get a true id, but my guess would be a Ferocactus.

#5 At first one would think a Mammillaria, but I'm thinking maybe a Coryphantha. Check out Coryphantha macromeris.

Pic found in Google images;).

Kara, unfortunately I did not see the flowers of anyone. For the first, the seller said it had yellow flowers. Except for no. 3. which looks endeed like Parodia Scopa (and I neglected it for a few... 6 or more years in a 5.5 cm pot), all the others I bought a month ago.

Your identifications seem great to me. Thank you!

When they bloom take pics and put them up;).

I wonder if #1 is a Parodia. The seed pods look just like my Parodia seed pods. I know other cactus have similar seed pods, but when I saw them I thought "hmm looks a lot like my Parodia seed pods.". Plus Parodia have many species with yellow flowers. Most Parodias I own have yellow flowers. Oh well time will tell;). You should get flowers this year, so you won't have to wait for long;).

I wonder if #1 is a variety of Parodia mammulosa?

Yes, Kara, that’s what I think for #1.