My adviser is leaving: what should I do as a third year PhD student
This situation is very tricky.
It is not clear what is your current school level, if it is above or bellow NW (NW is a decent school, but definitely it is not say UChicago).

If you are from a bellow or comparable level school, I would say to go ahead with the transition.
The tricky thing in that case would be to get admitted, but as your advisor is moving, unless it is a very steep jump in the rankings, most likely she can talk to the co-director of graduate studies and pull some strings.

Regarding the formalities you've mentioned, in my experience, there is a big difference between plain rules and practice. There is a chance that NW thinks highly about itself and formally will not recognize your credits (again, it depends where you are moving from). Nevertheless, you might get over that. Most likely your studies will be expedited, you will be able to retake generals pretty soon (if you even have to, maybe they'll just setup some committee and you just need to give a short talk to present your research, as you already do research probably), so nothing will be from scratch and your studies haven't been in vein, the nullstellensatz is the same no matter if you are in Chicago,Boston,Princeton or Palo Alto. and you will have the bonus of graduating from a better school, which is a huge aspect if you are considering academic career. GRE should be extremly easy for you by now, and if you are able to do proper TA job, you should pass Toefl as well, those should be very minor problems from your perspective (how did you got into grad school without GRE?).

If you are from a better school, this is kind of tricky, maybe there's another prof. who can step in to be co-advisor? You are only 3rd year, not  4-5th year, probably you invested less than 1.5 years in research so far. Being stuck without funding for the next 3 years (either way, NW or staying, you will probably finish up in 6 years, maybe even 7, sorry about that, not fair, but that's life, this is a major implication over your studies).
I don't understand your relationship statement, you didn't know that your advisor is considering moving? The application process for academic jobs usually takes more than 6 months 

All those hopes about your advisor funding in NW is wishful thinking that can have really disastrous outcomes. The startup grant most universities give to incoming profs are usually for moving, setting up office and some research visits, those are not in the level of NSF grants (or better - Simons/ERC). NW is located on Evanston - affluent suburb of Chicago, cost of living there is not zero (that's not CAL, but still not epsilon, 1 bdroom can easily go for $1800+), and considering this moving situation, are you sure she will commit so much of her own funds to bring you over? I guess support from your home institution is not a possibility as this are planning on terminating your TAship. Regarding the other funding options, some NSF stuff have strings attached (RTG, or supporting only students from their own school), it is too risky to consider this, especially as you are speaking about events which are probably 2 years into the future.

I believe that for the time being, you should continue with the application procedure to NW and if and when the admittance letter will come, set up a conference call between NW director of grad studies, your advisor and yourself and try to setup a better plan for you. If NW decide not to admit you, all those talks are hypothetical, either you will have funding for a visit or not, but you will need to stay in your own school.