Fellow non-native-american-pilot speaking here:
You will need to get your transcripts validated showing you have a at least the equivalent of an American 4 year college degree, a LOT of applicants have advanced degrees, I was not very successful, according to the accrediting agency I only had high school, even tough I have six year prep school of higher level than US high School and 3 years of University.
You will need at least 1000 hours turbine PIC, and you count only hours where you were the Captain in the airplane (not PICUS or SIM).
As mentioned already you will be up against several thousand applicants who have lived in the states for the last 10 years, making the required background check just that much easier, most airlines will accept green cards, (except I think Fedex UPS and some other due to dept of defense contracts) but I am sure it doesn't help....
A lot of airlines will give preference to people with internal recommendation letters, I would guess because you post this question you don't' have any?
As for regional pay:
when I got laid of from my last job I didn't even apply to any of the regionals, I can not afford to go to work for what they pay!
GoJet Airlines | AirlinePilotCentral.com
You are looking at working around 18 days a month starting at around $20-25000 first year, around $35000 second year. if you are lucky and pick the right regional that gets the big contract (at the expense of the other regionals, see Envoy) you will upgrade after 4 years or so, get back on reserve, get displaced to LGA and sit on reserve trying to get your 1000 hours.
IF you get hired at a mayor (or regional) you will be at the bottom of the list and get the most junior aircraft and base regardless of previous experience.
The good news: there is no need to pay for a type rating, you flew your hours while qualified on that type so you can use those on your application. you can go to
https://atpflightschool.com/index.html
I think 12 days $10.000 should get you an ATP, if you insist $12.000 and 12 days will add 737 rating.
Most of this has been posted by others already, just confirming the state of things here, also I think my post sounds bitter, but I found a good job here, looking at a reasonable pay, schedule and upgrade time on an Airbus, and I love flying, would not want to do any thing else for a living, it is just not an easy thing to get on with a mayor. I have a US passport, 5 type ratings, JAA and FAA license, 3000 hours of turbine PIC, 4000 hours jet, 8000 hours turbine and more than 10 internal recommendations for American and United (no College degree though), have gone to 3 job fairs and zero interest from the legacy carriers.
Good Luck with with the job hunt!!
P.S. If you get a green card you will have to establish residency in the States otherwise you can lose your green card. Way back in the day I lived in the States while working in Europe for a few years and couldn't claim I wasn't a US resident for tax purposes because that would make me ineligible to keep my card!