Aloe Juvenna "tiger tooth"
Cylindropuntia spp "mini cholla"
not too sure about the last ones id...?
The last one looks like a chamaecereus of some kind.
Here is a link that might be useful: http://www.mattslandscape.com/chamaecereus/
They look OK, except the mix needs a lot more perlite, still looks too peaty. I'd pot them a bit smaller too OR don't pot them up 'til they get quite crowded.
I don't grow Cactus, but suggest in the last pic, that pots rather large, requires cautious watering.
If you flip over to the 'Discussion' side, that's where Conversations are ongoing.
Since your profile mentions you're here to learn (good for you!!) go on over to the Discussion side: there's lots to read & you can browse the FAQs as well. Most educational.
This Gallery is rarely used now, since we can insert pix into posts so readily. Just so you know.
I think they look great! How fun to have new babies to watch grow and flourish! Enjoy!
Also I have quite a few of the last picture you posted.
"Peanut Cactus" Here is a picture of one of my blooms! So pretty! I do notice that they like more water then most cactus plants. Enjoy!