Yes you can screw in the swivel stud on a synthetic stock. Just use the same tools, and care, you would if the stock was wood. A variable speed drill, slow of course, and a 'step' drill bit. My 760 carbine came with the fore-end adaptor for a sling, but I also have/like the split barrel band from Uncle Mikes on my other 760 rifles.
My 760 carbine '06 shoots Rem. 150 PSP, 165 PSP, 180 PSP, and 180 SP Core-Lokts very well. So well that I don't have to adjust my scope at all. In the 40+ yrs I've been hunting deer I've never had a rifle that could shoot 4 different loads so well, and not have to make a scope adjustment. Rifles like this a few and far between. Sometimes you just get lucky.
I'd try the Core-Lokts first simply because they are plenty good for whitetail / black bear etc. and you can find them just about everywhere, especially in .30-06.