Yikes. Two 16 year olds sharing the responsibility of a dog.
My boyfriend and I are both 20, in college, and live on our own and sometimes I feel we weren't ready when we got Franklin. Especially after Franklin got parvo and we had to shell out $1600 and have him hospitalized for 7 days. Which brings me to my next point - I would definitely recommend Brady stay at your home (inside) until he has gotten ALL of his shots, which may mean until he is 13 or 14 weeks old. I may sound a little over cautious but I know first hand what can happen to these little ones. We never took Franklin outside and he
still got parvo, even after being vaccinated for it. With this puppy being so tiny and not having his vaccinations, I just wouldn't risk it.
If you search crate training on YT or even just on Google, you'll find plenty of helpful information. The wire crate is a perfect idea - you just need to do it correctly. I would use the divider to make the area as small as possible. Puppies have a natural instinct to not soil their sleeping area. Unless you plan on trying to potty pad train him, go ahead and do away with it. He will bark and whine and cry like he's being tortured but you can't give in - he needs to learn that the crate is a safe and comfortable place for him. Covering it with a blanket or towel will help make it feel more secure, like a den. A 7 week old puppy needs to be fed multiple times a day, to prevent hypoglycemia. I would, however, restrict food and water about an hour before bedtime to reduce the chance of an accident during the night. The general rule of thumb is a puppy can hold it's bladder as many hours as it's age. So, 8 weeks = 2 hours, max. That means you'll be getting up a LOT to let him out at night.
The pen you bought is a great one - I have one just like it. BUT! Like I said, unless you're planning on pad training him.. it may be easier to keep him in the crate until he can learn to hold it a little bit longer. I would save the pen until he's older and needs more room.
You may end up having behavioral and training problems with this puppy. Because he was taken away at such a young age, he missed out on crucial socialization and training from his mother and littermates, such as bite inhibition and potty habits. He can grow up to be a perfectly well behaved dog though, you'll just have to work a little bit harder.
It sounds like you're going to end up becoming the primary caregiver for Brady, which is fine because you'll fall in love in no time (sounds like you already have!). YorkieTalk is a wonderful place and I hope you find a lot of useful information here. Feel free to ask us any additional questions, or you can even PM me if you want.
Welcome and good luck! Post pictures soon.
Edit: I forgot to mention that I don't think switching homes every week or so is the best idea. The puppy will never fully get accustomed to either place and will most likely be traumatized by being moved back and forth. I also think the instability will cause additional challenges in behavior and training. Franklin is potty trained at my apartment and no where else. I think it's best that the puppy stay in one place (preferably yours, since it sounds like Brady is in good hands). Your son's girlfriend can visit.