This is somewhat of a broad question but lets say I'm putting together a hunting rifle of the AR variant that I want to use on deer, hogs, yotes, maybe even smaller elk at 100-400 yard ranges?  Assume that we are only taking body-shots on these game.  Which combinations would be less or more effective and pros and cons of each?

8-9'' 300 Blackout

12.5'' 6.5 / 6.8

14.5/16'' .308

and finally a 14.5'' 223 with heavier grain bullets

My assumptions are that the 14.5/16 .308 is going to have the most energy output at the greatest weight of bullet and barrel.  Would the 12.5'' 6.5 be close to the 308's energy output?  Is the 8''-9'' 300 Blackout enough to drop medium sized 4 legged game at longer ranges?  Would the Blackout or 223 leave 'wounded runners'?

What other factors and considerations would come to mind?

Its not on your list but I'm using .243 win with great success on hogs, deer and yotes.  Don't have elk so don't know about that.  Alot of people post in the hunting forum's that .223 with the heavier hunting bullets are gtg but that seems a little light to me.  I tried .223 on hogs and was not impressed but some folks seem to be happy with it..

Of those the .308 will be the best choice followed by the 6.5/6.8. The Blackout is a 200yds and in option. .223/5.56 is ok for deer/hogs/coyotes with proper placemat  but a no go for elk.

I would hope you have already checked it out but some states/places don't allow hunting with SBRs.

Elk out to 400 yards on your list, I'd say the .308 is your best choice, but I would go longer than 16".

I agree with the other posters about 308 being the bare minimum if you are wanting to shoot at elk out to 400 yards. I know people have done it with 6.5 grendel or 6.8 spc but in my opinion is that its one of those just because you can, doesn't mean you should. Elk are pretty tough animals and like the poster above said, he put 2 shots into an elks heart at 400 yards and the thing still managed to go several hundred yards before it knew it was dead.

I've played with the 6.5 grendel and looked long and hard at the 6.8, 450 Bushmaster, 458 Socom and 50 Beowulf. I have decided for the type of hunting I do here in CO an 18" 308 AR is going to be my next purchase based upon the 308's versatility, power and availability.

Did you throw the rounds at him? People hunt much larger game than elk with .308 and it knocks them on their asses...
I think you made some bad shots. The amount of damage a .308 round would do to any large game's heart would be devastating and would have dropped the elk within a few steps, if not immediately.

Meh,  animals are not people.  I blasted a cow elk at about 25 yards with a .270 years back.  No lungs or heart left and it still ran about a hundred yards.
Then again I have shot a nice bull at the sameish distance and he just laid down.

There is all types.  I suppose that is why people want KO style cartridges for dangerous game,  enough bullet to physically disable the animal.

.358 Winchester a good option if ranges are kept realistic? Not a huge edge on the .308 in terms of energy oranges to oranges, but the increased diameter one would think would be useful.

At least in terms of an AR, keeping the 308 as the parent should really simplify things.

+1 same here.

I shot a 200+ lb whitetail buck in the boiler room at 125 yards with a 150gr 30-06 and it took off like I missed it. It ran wide open 150 yards straight into a stand of pine saplings and I heard a crash. When we got there it had run head on into a tree apparently and just crumpled in place. When we gutted it it had no heart or innards - just a mass of red jelly - with an exit wound the size of a fist - yet it ran that 150 yard dash wfo dead on its feet.

Another time I shot a large fork horn buck with that same gun and load at under 50 yards and it just fell drt flat down like a sack of hammers - like cutting off a light switch. The bullet just punched a laser beam hole directly through both scapulas and the heart with very little expansion or shock effects - maybe nickel-sized exit wound. Go figure.

Drt every time on elk at 400 meters?  35 Whelen and above maybe, I dunno. YMMV. Just my 2c.

308
You also have to consider the bullet you're going to use.
The few rounds I like for their performance design are Berger VLD's, Accubond,  and TSX.

for 300 BLK you might want to check out 300BlkTalk.com
I'm still learning more about the 300 BLK; from your want list you might consider a 16" bbl and select good hunting bullets for deer and hogs to cover the longer distances (and that's probably stretching it; closer is better for the BLK).
They seem to have very good results using 125 gr. SSTs, Nosler BTs, Barnes bullets, Lehigh Defense bullets.

.308 is king though of your choices; I'd prefer a 18"-20" bbl.

5.56 is great for coyote and iirc they have good results for pig eradication (usually head /ear shots).
Not a big fan of using 5.56 or .243 for game hunting personally.

For deer most anything will do, but I would say a 6.8 spc ultra lightweight ar15 at about 6# is a nice rig.
Nice for hogs too.

I just bought a 338 federal lite hunter 90% for hogs, but nice for deer too as it really DRTs deer.

If you set up a 20" bbl and a hot load 160 gr TTSX you have a nice flat shooting slammer deer and hog rifle.
Also Federal sells Federal American Eagle .338 Federal 185gr SP 20rds for about $18 a box.

A 185 gr TTSX in 338 fed out of a 20" bbl at 2650 if you can get that, would be a decent 300 yard round for elk, but its dumping at 400 yards.

Any of above is weak for Elk at 400 yards so If I were seriously hunting elk with a AR10, I would do one of two things personally.

Buy a 20-22" barreled 338 federal and have the chamber reamed to 338-284. It will clean up with a simple ream only and that gets you very close to a 338-06 but
you can always download it. I have a 19" savage 338-284 barrel and sold it to a guy in colo who used it to kill a 850# elk at about 100 yards. Said it took two steps and tipped over DRT. His load was 185 gr TTSX at around 2850. Going to a 338-284 will buy you about 150-200 fps depending.

Better yet would be to get someone to set up a 338 RCM barrel and bolt. Marty built one a while back. Not quite a 338 win mag but not far off either as far as killing power. Probably somewhere between a 338-06 and a 338 wm. A 338 fed barrel will most likely clean up reamed to a 338 RCM. On paper it looks good anyway.