Aye.
Yep, no good if the dog is constantly pulling, as you often see. That isn't what you want. Keep the choke chain slack and when he does pull a sharp jerk to pull him back but straight away the chain is slack again. He'll soon get the message that pulling is not a Good Idea and that walking nicely with a slack chain is. You may feel you are constantly jerking the chain to begin with but dogs are not stupid and will learn.
Yes, he's doing his very best to please and is showing he really wants to be with you. Grovelling springs to mind.
Good idea decide now any areas that are no-go for the dog and stick with them. With all the other stuff, the lead and overly pleased, don't be to strict on keeping him out. But make sure that he his never allowed to stay in a no-go area, shoo him out, firmly if required, I wouldn't shut a door on him at this stage but just repeatedly take him out of the no-go areas. When you do emerge, make a fuss of him, as a reward for waiting outside.
A week also strikes me as too short for anything but the most placid and happy of animals to settle. I'd expect to see some improvement in a week but not a change to the well trained and obedient member of the family that one is aiming for.