I’ll just add to the chorus and suggest with Northeastern’s coop approach may be super cool for some kids, other kids don’t necessarily love it. And generally, I agree merely attending Northeastern will not give you an advantage for post-college opportunities over people who attend flagship-level publics like Binghamton, Rutgers, or Pitt.
In these sorts of cases, by far what will dominate your post-college opportunities will be whether you find an academic program in which you can do very well (could end up setting you up for med school, could end up something completely different), and then networking and such. In ways both good and bad, your fate is really still in your own hands, and the college you choose has far less to say in the long run than some people hope/fear.
Given that, I would at least seriously consider Rutgers Honors, because at least some people find such programs help them make a successful transition to college. And even if it ends up not being a big thing for you, in this case Rutgers still has all the opportunities you need anyway.
I would also personally see the London thing as a net negative. It is tough because I think a study abroad experience can be great, I just think the timing of starting that way is really unfortunate.
Binghamton and Pitt, though, are perfectly reasonable choices too.
Pitt is a good all-around university of substantial but not overwhelming size, and it has real sports. But I think the main appeal to Pitt for OOS students is usually that it has a great location in a fun, attractive city. There is a pretty short list of peer colleges meeting that description (not none, but not a lot). I note of course Boston is a huge part of why Northeastern has become such a popular application, but again they don’t have the capacity in Boston for you right away, so . . . .
And of course Binghamton is a totally different setting, one a bit more typical for a flagship level public in that it is centrally located in the state but not in one of the major cities. But lots of people like that sort of traditional college experience, and they have a cool residential structure as well.
So I don’t think you can go wrong here, holding aside costs it is just a matter of personal taste and what you hope to be part of your college experience. If it were me, I would probably be choosing between Rutgers and Pitt based on what I thought of Honors versus being in a fun city, but again all four would have their pros.