Yes, Catalina supports the 2012 Mac mini. 4K playback in the TV app and Sidecar (except for a lower-quality H.264 version enabled via Terminal) are not supported though.
I love my 2012 QC i7. Runs Mojave like a champ with two internal SSDs. One word of caution... mine (eBay 2016) came with the original fusion drive. The drive started giving I/o errors and I was lucky to be taking regular backups. I forget how apple determine the hd was the cause, but it wasn't obvious. Regardless, I would say consider swapping at least one real SSD in there - and enjoy!
It will probably not be supported by the successor of Catalina. So, I'd try not to overpay.
I've got a 2012 i7 myself, but I'll wait until 10.15.1 at least. There's also the issue of the removal of Dashboard....
Without an SSD, they're not fun, though.