1. Under the current Administration the atmosphere is not very Immigrant friendly.
Nothing pilot related will get you a Greencard.
2. No. There is a difference between minimum requirements and experience to be competitive for the job. Most of the Majors own one or more Regional airlines and the current flow for American Airlines is around 7 years at their Regional. If you translate this to hours you’re around the 5000 hr mark of which 3500 Jet of which probably 1500 PIC.
There is no reason for them to hire from outside with less experience.
3. Before, can a US pilot apply at Lufthansa without an EASA license? Why would you expect it the other way around?
4. No, flying single engine piston in Arizona is incomparable with flying a Regional jet in and out of Newark, JFK, Chicago, Miami.
5. Yes. Market determines demand which determines hiring
I have some trouble understanding your motivations. Your re currently in a position that thousands of aspiring people want to be in.
Yet you are romanticizing flying for a US Major.
You’ll just as easily be right seat for 7-12 years flying only US Domestic city pairs in a 320 or 737.
If you are interested in long haul flying look at CargoLux or AeroLogic