The mini computer is a decent idea but you have to make sure the onboard graphics will power your external monitor. This is important because the mini will have limited upgradeability if any.
For monitor I recommend FHD (1920X1080) which ought to be doable with a stock mini. Don't let anyone talk you into 4k because the computer you're shopping for won't be able to power it.
Finally getting around to the OP's request, if you're set on WMC then win7 isprobably your best option ALTHOUGH you must verify that you'll be able to get the install disk to boot on your mini. If the computer comes with win7 installed that would be ideal. If your ideal computer hardware comes with win 8.1 installed then I would go with that.
I've run both and the primary shortcomings of Win 8.1 have to do with configurability and the stupid charms bar. Otherwise 8.1 is actually better than 7 in many important respects.
The people recommending other OSes aren't paying attention to your primary request: You want to run a tuner and/or cablecard.