Notocactus uebelmannianus is what you have there, although some call it a Parodia - its flower colour is unique among the Notos. A nice plant there, Sam. And, FWIW and with an unlimited bankroll, you should always buy plants that you like which are without a name tag - it's usually most excellent.

Looks like my c. peruvianus, which I might add has just started to show signs of its first bud. Mine was a broken arm off a massive plant that was a few miles from me - asked the owner what kind of plant it was and she gave me a 6-foot section! :)

Yeah I was about to mention coffee filter; but it will probably eventually fall apart (?), gets bit green from water and perhaps is bit slower draining. Not absolutely sure thou and I have used them too when first started. But if that's what you have handy, use it. You may have to repot plant in couple of years when plant grows bigger and replace it... Removing all old sol is very much recommended. Surrounding old with new will eventually create problems. If the soil got very dry, it may be more difficult, and you could wash it off by using warm water and swishing the root ball in it. Pick off soil using blunt chopstick if needed. It may just crumble off. Should you need to wash it, put plant on a kitchen paper towels to wick off extra water. Pot the plant, do not water for few days if the roots were wet. Keep in lower light just for few days, and then move into more and more light - best you can get. Btw, be careful as euphorbias leak white sap f injured (some other plants doo too) - latex - that some ppl are allergic to. You may not be, but make sure it doesn't get into your eyes, as it could be very painful for anyone.

The top one looks like Echeveria runyonii 'Topsy Turvy'