Semi-auto's require FL resizing every time. Other than that, loading for a semi-auto is no different than any other rifle. Case length can be critical, but no moreso than for a bolt action. Trim all your brass to one length, chamfer and de-burr just like you do for any other rifle and your ready to charge and seat the bullet.
"...for 7.62x51 which don't seem to exist..." They don't. I use RCBS (they may be Redding though) .308 Win dies myself. My rifle is a real semi'd M14. Any good die set will do nicely as long as the sizer is an FL die.
If you're using milsurp brass, the primer crimp must be removed first and you must reduce the charge by 10% due to the slightly thicker brass. I suspect you know that though.
Use a 165 grain hunting bullet or 168(up to 600 yards) and/or 175(past 600 yards) grain match bullets with regular large rifle primers(CCI #34 primers are a marketing gimmick) using .308 Win data. Work up a load using IMR4064 or Varget(there other powders, of course, but these tend to give the best and most consistent accuracy) and go shooting. 150's will do with the same powders, but the .308 prefers the heavier bullet.
Headspace has nothing to do with the case. Cartridges do not have headspace.
Slamfires are caused by improperly seated primers, not the rifle. Even with an M1 or M14.