Cia, may I suggest, if I haven't already, that you check out the Behaviour Section. We have two threads in there devoted to Housetraining and Nipping/Biting. Your BF has the right idea actually when it comes to mouthing. He is teaching her what is acceptable and what is not. Since pups explore their world with their mouth, it's great that she is already learning to be gentle with human hands.
Puppy pads are generally not recommended for two reasons. They are a choking hazard if the pup is left alone with one in a playpen, and it means you are trying to teach the dog to do two things, rather than one. As you noticed, your pup went for the door. That was perfect and yet you insisted she go back and try to use the Pad. If I'm not mistaken, you live in an apartment. I know it can be difficult to rush a pup outside from an apartment, but since she seems willing, it would be easier on everyone if you ran out of the door with her. It's less confusing for her, but means more work for you in the early stages.
Dogs are not Cats and generally prefer to do their business outside. You really do need to find a better way of securing your pup than an upside down laundry basket! Crates do not have to be expensive and should be the right size. You may wish to buy one that has a divider so that it can grow with her. Keeping your pup safe is worth every penny.
May I also suggest that you whisk the puppy outside after eating, drinking, playing, sleeping or a 20 minute lapse. Praise, praise, praise when she does her business where you want. You must completely ignore the mistakes!!!! They are YOUR
mistakes. Don't bother saying a word if you catch her "in the act" as you have failed to pick up on the cues.
I'm sure once Brian realizes that he is doing something right (the mouth playing) he will be more cooperative with the housetraining. It's easier if there are 2 pairs of eyes always watching, rather than one, but consistency is key.
Jonesy sounds like a smart one already. Take comfort knowing that all the hard work you put in during these early days pays off by having a lifetime with your well behaved, "go anywhere with you" dog.
You are on the right track, so keep up the good work!