Well first, without a green card there is no way you can be hired so you'll have to work that out to get a job

1. There are a couple more steps. I'm guessing you hold some sort of CAA/JAR/EASA commercial or ATP license. Normally to get a FAA ATP you have to hold a FAA commercial. If you hold a foreign commercial with the required time or a foreign ATP you can step straight to the FAA ATP. You have to have your CAA verify your license to the FAA to do this. And you have to pass a TSA background check before you start any flight training for the ATP checkride. The main problem is you have to complete an ATP Certificate Training Program before you can take (sit) the ATP written. (The FAA does not issue pilot licenses. They certify you as a pilot and therefore you get a certificate, not a license.) The ATP CTP was designed for pilots coming out of CE-172s and going to RJs. For pilots coming out of the military or foreign pilots getting an ATP 95% of it is a waste of time. A year after the requirement went into effect there are only about half a dozen authorized providers of CTPs. They run about a week and cost about $5000.

2. Once you have the written out of the way there are hundreds flight schools were you could train for the ATP checkride. Something is lost in translation with "apps on offer."

3. I'd think you could get in at a major. But yes, you have to apply and that starts out on line now.

4. Regionals fly some of the same routes as majors, just smaller equipment and paychecks.

5. The only time you'd find a DEC job here is for a start-up company. EVERYTHING is done by seniority based on your date of hire.