Yes, the 6.8 upper will work on a 5.56 lower. However, 6.8 uses a different bolt (not sure of the bolt carrier) and obviously, a different barrel and mags.
I have a 6.8 LWRC 10.3" gas piston upper which fits on a LMT complete lower perfectly. I'm using CProducts 25 round mags ($26 each).
From my limited understanding, 6.8 was really designed to be more effective than 5.56 for ranges 300M and less. 12-18" bbls seem to be considered the best in terms of best fragmention.....I can only assume 16 and 18" bbls are "best" past 300 yds.
For what you seem to be wanting (that is something good past 300 yds; other than .308), I'd consider 6.5 Grendel instead. Supposedly, it has better performance than 6.8 past 300M and can reach out to 1000 according to the Grendel supporters. The only downside I see to 6.5G is the current lack of ammunition (Wolf is supposed to start producing 6.5G soon.....but how accurate will it be for distance shooting), price of really high quality ammo, and the proprietary nature of the round (Alexander Arms seems to have a tight grip on it).
If 6.5G ammo and uppers were as widely available as 6.8 Iand it seems to be growing) it could end up surpassing 6.8 in popularity.
It's all a gamble. I gambled with 6.8 and hope it works out.....if not, I'll take up reloading I suppose.
Sean