There seems to be a lot of speculation and misinformation on here with regard to visas. Try going to the USCIS website - all the correct and up to date visa info is on there, as well as info on the green card lottery and who can or cannot apply for it each year. It will also give you accurate info what is required for nurses to work here.
I doubt very much you will get a flying job with an H1 visa (been there, tried that). It is almost impossible to prove that no one else can do the piloting job for which you would be applying, based on the fact that there's an awful lot of American pilots trying for that job too. Now, should your girlfriend get a job with an H1 visa, and you marry her, you will be allowed to stay in the country, but you still will not be allowed to work. The advantage of the H1 visa is that the two of you can start the process of applying for a green card while you are living there and she is earning. On average, that takes about 3 years. Our work authorization card was issued 2.5 years after the process was started and the green cards arrived a year later. If she is working and you are learning to fly, around the time the work permit arrives, you should be all set to start looking for work... in a perfect world
You can get a flying job with a green card (been there, doing that too), but there's a good chance the employers will take an American over you if there's more than one equally qualified applicant. Also, be aware that a lot of the airlines like their pilots to have a university degree at the moment.
Try the website for ALPA (US version of BALPA) to see if they give an idea of salaries for the airlines. To give you an example, I met a charter pilot the other day who flies a Piper Navajo. He had resigned from American Eagle after 7 years, mostly as an FO, with a short time as captain, to take this job. His lifestyle had improved 100%, no commuting, most nights at home and, so long as he was getting a minimum of two trips a week he was breaking even on his previous salary. The salaries for charter pilots are not usually anything to get excited about.
The good thing about flying here is the variety of opportunities that are available. Whether you want to fly for the airlines, fly charter, fly corporate, ferry aircraft, fly medical flights, fly float planes, bush flying, instruct, fly for the DNR (count wolves, bears and forest fires), fly water bombers, fly freight, fly warbirds, etc, etc if you put your mind to it, you should be able to do it.
If you look at the visa section of
http://www.british-expats.com/ you will find a lot of informed advice on there, not speculation based on goodness only knows what.