ABD...
I'm in just the same position... 1000hrs TT, frozen JAA ATPL, 737 rating, and laid off by a major carrier in the UK.
I'm also looking into moving to the States, as I have a green card, and had just the same queries as you.
Coincidentally, I grew up in Zurich and spent the last 20 years there - almost ended up with Swissair, too. Looks like I would have been chopped no matter where I ended up!
Thanks for all the help on US licensing, and good luck with applications!