I have a late 2012 Mac Mini, but I don't use it for games. However, the machine initially came with a slow, 1 TB 5400 rpm hard drive. I replaced it with a much faster Samsung 840 Pro 256 gig SSD, and my "need for speed" was (and still is) easily satisfied. So, from an "external" point of view, you might want to have the SSD inside the machine when you purchase it.
Yes, the current Mac mini exceeds the minimum requirements for The Sims 4 and should support the game. The Mac mini does not have an optical drive, so you would need to purchase an Apple USB SuperDrive to install The Sims from the retail disc...not needed for the download version.