One correction to your post - the MK 108 was not derived from the MK 101, which used an entirely different mechanism. It would be more accurate to describe it as a scaled-up MG-FF.
The barrel was short partly because the cartridges contained very little propellant, which burned out after the projectile had only travelled a short distance up the barrel. The more propellant you have, the longer the barrel you need to get the most out of it. In the photo below you can see the huge different in size between the cartridges for the high-velocity MK 101/103 (30x184B) and the low-velocity MK 108 (30x90RB)
Another reason for the barrel being as short as it was concerned the type of operating mechanism. The breech was not locked to the barrel at any time - when the gun fired, the breechblock was moving forwards, and the recoil first stopped it then pushed it back again. The shell had to be clear of the barrel before the cartridge case left the chamber, otherwise the case would burst and the plane's structure around the breech would be blasted with burning propellant. So the barrel had to be kept short to ensure that this did not happen. There is an obvious relationship here between rate of fire and barrel length: a heavier breechblock would move more slowly, giving more time for the shell to leave the barrel, so the barrel could be longer, but the Luftwaffe put rate of fire above muzzle velocity in their priority order for this gun.
As for your second question, there was a blast tube attached to the muzzle which ran between the engine blocks and up to the prop, otherwise the engine compartment would have been filled with gasses and soot from the muzzle blast. It didn't affect the velocity.
The Germans were in such desperate straits at the end of the war that they might well have considered using the MK 108 in the AA role, I don't know, but they would have made very bad AA guns - you need high velocity for that role. I agree that they could have made useful infantry weapons, but I don't know that that occured to anyone.
Tony Williams: Military gun and ammunition
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