Why does a puppy get free roam in the house? He is a dog - he needs rules, boundaries, down time, and a strict schedule. This dog needs to be crate trained, and secured so you can see to your normal routine. He sounds like a typical pup that is allowed to have too much freedom and does not have his needs met properly.
In regards to the size: keep him lean and feed the minimum needed for proper growth and development. He comes from oversized lines so you have to be very careful.
Barking and play biting are not tantrums - it is normal GSD pup behavior. Redirect, and shape the behavior. If you saw a true tantrum and acting out...you would realize that this is nothing, but puppy habits. Puppy BITING!!! Teaching Bite Inhibition
Stop the shaking cans, alpha rolls, and correction for NORMAL behavior. You are creating a reactive, pushy, and frustrated dog. Up his exercise, stimulate him mentally with games, increase training sessions, get the GF involved in training, socialize him etc. Putting him in the yard is not enough - he needs active engagement and activity. Contain the animal when he is not actively engaged with you, formulate a strict schedule, and stick to it. Dogs like routine, and consistency. Personally, I believe it is harmful to use correction devices before 6 months old - you are creating problems for the future.