With a 10.5" barrel, a 12" target at 80 yards should be gravy!
I have a 16" upper with a 1-4x scope on it, and an 11.5" upper with an Eotech & BUIS on it. Both shoot 6-7" groups at 100 metres, but with an optic on the 11.5" upper it will print 3-4" groups.
With that same set up; 11.5" barrel, eotech, I was able to hit a human torso sized steel gong 600 metres away with 8/10 shots, once I figured out a point of aim. The POA turned out to be around the bottom of the Eotech ring, so 30minutes, ish.
The main DISadvantage to running a shorter barrel is a loss of velocity. As a general rule, with the AR platform you lose 100FPS for every 1inch you remove from the a 20" barrel. Since you don't need to worry about terminal ballistics, this really only matters for your trajectory. If you are planning to get into competition, this will only matter when you're making shots out to 300 metres. I have my 11.5" zeroed about 3" high at 100 metres. Like I said, it shoots about 6moa, so I just centre the group on the top of my aiming point. With this zero, I still hit the kill-zone of a shillouhette target out to 300 metres, and by that time, I start missing wide anyways.
I personally don't see any advantage to a 7.5" barrel, except for fun. I'm not sure, but I think the reason for the 7.5" barrel is so that it can be classed as a pistol. Again, not sure.
For competition or CQB, the 10.5/11.5" barrels are the same in my book. I prefer to use the short barrel because of how it balances. I use a quad-rail system and a tango down grip, mounted about 4/5ths of the way up the rail (almost at the front sight). I like to have my hand that far forward so that I can swing onto my target & stop anywhere I want. Also, it moves the center of gravity well back, to around the mag housing. With a vortex flash suppressor on the barrel I can fire it with almost no muzzle jump. Sure, there's recoil, but it's almost straight back, and that allows lighting quick follow up shots. Since going with that set up my present time & follow up shot time have shrunk considerably.
As far as building vs. buying, there's NO reason not to build! Even if 11.5/10.5" barrels aren't available, a 14.5" or 16" barrel with a carbine length gas system can be cut down. I believe that the 11.5" barrel on my short upper just uses an un-modified carbine length gas system. The only thing you have to look out for if cutting down an AR barrel is if it's chrome-lined. I was told that you can "chip" the chrome lining when cutting down a chrome lined barrel. If that happens you'll have to cut more length off the barrel, and hope it doesn't happen again.
As for recomendations, whatever floats your boat! I like LMT & RRA, There's no reason to spend more money than that, because there will be no discernable change in quality until you get into the HK realm.
Hope this helps.
Cheers.
Tim H