I like to get rings as low as I can. I'd strongly suggest NOT getting see thru type. Sounds great in the store, but it sets the scope too high for good shooting and they're worthless in the woods
Ring height may vary depending on the rifle, even with the same scope. The limiting factor is the height of the receiver above the barrel and the barrel contour. These determine the clearance between the objective bell and the top of the barrel. One brand of rifle may need a low mount, and the next brand may take a medium for the same scope.
IIRC, Burris has Posi-lock rings that have a floating "gripper". This lets the rings adjust and makes up for slight mis-alignment
Leupold makes some great quick detach bases, which is probably what I'd get. Don't know if Burris makes a QD.
Until I started working in the gun shop, I had no idea how complicated something as seemingly simple as scope mounting could be. Ring diameter, base make and height, QD or solid, two-piece or one-piece bases, etc. all come into play
Don't forget the Butler Creek caps (and no, their chart is NOT always correct). You'll have to go to a stocking dealer and test fit them.