Absolutely no loc-tite on the barrel nut.
As long as parts are in spec, and castle nut is torqued to spec, it's not needed on the castle nut or buffer tube, either. Staking it is a good insurance policy.
Not needed on the pistol grip screw, so long as you used the proper lock washer (should have come with grip/LPK).
Not needed on trigger guard allen screw, either.
Anything else that screws in (ie: rail systems, QD sling adapter on your stock, etc) should come with factory thread locker on any screws that require it. For example, the 6 allen bolts for the Daniel Defense RIS II rails that bolt the rail to the backplate come with blue thread lock on them, but the 4 screws attaching the bottom rail do not.
Never argue with an idiot. They will drag you down to their level and beat you with experience.