Just go on Microsofts website and make a USB installer for win 8.1 and it will work fine and you won't even have to put in a product code. It's embedded in the mobo of a win 8 or newer.
Unless you are wanting the bloatware version of windows 8.1 LOL. I'd rather have a clean version personally.
I've never done a recovery from a partition on a hard drive. I would think you have to set that drive as your boot device or force boot that drive from in the bios so it loads the partition and go from there.
I always make USB installers from Microsofts site. I have one for Win 7 Home, Pro, Win 8.1 Home, Win 10 Home and Pro same one. You just need a 4 or 8 gig USB which is $4 at Microcenter or a cheap one wherever you can get them cheap.
I also have the MSI GT72, but the 6QD ( 6820HK & 970m ) , what a great machine!! I love it and it's treating me very well in everything I do.
Yea the recovery partition is really only useful if you are wanting the way the laptop came, aka with all the bloat on it.
With most laptops for windows 8 and up, the CD key is embedded right into the bios.
So you can either download the windows 8 iso, and install that, or download the windows 10 iso and install that.
Both can be downloaded from the microsoft website.