Don't have all of the answers but here is a stab at some of them.
If started at any major airline, you would start at the bottom of the seniority list and most likely on the smaller equipment. Having said that I believe some new hires at Delta have gone directly to the Intl 767 but that is pretty rare. Being a new hire and not getting much flying (stick time), is probably not in your best interest.
As for the cost of a FAA conversion I believe you would need a full 737 or 777 course just as if you had never flown it before. Some training organizations offer a shortened type rating course and give some credit for previous experience. Regardless prepare for the worst. The 737 can probably be done for under $8,000 but the 777 would be significantly higher and in the range of $16,000. I'm sure others here can share their experience and costs associated with this training.
I know in the past that some airlines did honor the green card but not all of them did. 911 may have diminished that, I don't know.
Good luck