Not Legal Advice: CHECK WITH A LAWYER, DON'T RELY ON THIS POST! I AM NOT A LAWYER.
Don't kill the messenger here, but I have bad news for you:
1. YOU CANNOT BUILD ANYTHING FULL AUTO UNLESS YOU ARE AN FFL07.
Period. Doesn't matter. You cannot build the receiver for this gun... The only way to get a full auto weapon is to have an original receiver that was properly registered prior to the 1968 CGA or registered legally prior to under the 1986 act when manufactured.
2. If you want a gun to be a C&R gun, that does not exempt it from the 1934 NFA. So even so, you would have to have it registered on a form 4. You cannot build a new receiver for all the old parts. Yes, it sucks, but a C&R gun must be in it's original configuration with it's original parts. No getting around it, any new or replacement parts can bump it out of C&R status, although the ATF is the final arbiter of this.
3. If you want a Full Auto Firearm, it will cost you. You can legally purchase fully automatic firearms, but you will need to get the appropriate paperwork, find an affordable weapon (starting at $3k and up), and wait several months for the ATF to process your paperwork. Don't get caught trying to build your own fully automatic firearm, 10 years in jail and a $250k fine are amongst the sentancing options. <- This is a bad thing!!!
4. With regard to the barrel. If you build it without a stock, it is legally a pistol. Check your local and state laws before building ANYTHING. If you use the short barrel and the stock you will need to file a form 1 and register it as a short barreled rifle, which may not be legal in your state. The ATF will not approve any form 1 that would be illegal in your state. There is a $200 tax to build a short barrel rifle.