Math teaching job after PhD in the United States for a foreigner
Is the salary increasing from 1 to 3?


Generally, yes.


  This is what I would expect, but I heard that, for example, the salary of a high school teacher is higher than the salary of a no tenure track position/ lecturer even at a good university. Is this true?


The median 2012-2013 starting salary for US high school teachers was (ugh) $36,141.  This varies considerably based on the state: see here.  There is so much variation in the nature of a non-tenure track university position as to dissuade me from trying to look up figures, but in the cases I know of non-tenure track teaching positions at universities which are (i) full-time and (ii) with a PhD, their starting salaries are at least $45K.

In 2013 the median annual salary for US high school teachers was $55,360.  Teachers generally have strong unions and get steady annual raises, which is something that university professors have had a lot of trouble with in recent years, and non-tenure track faculty even more so.  Most people I know who are PhDs and have been in non-tenure track teaching positions for at least ten years are making $50K to $60K.  I would not be surprised if the median salary for these positions was less than $55K.

Note also that the average US high school teacher certainly does not have a PhD.  (In fact, fewer than one percent of such teachers have a PhD.  This is such a small percentage that the "converse percentage" of PhDs who go on to teach high school must be rather small as well.  The last link is to an article which explores this a bit.)  In the realm of high school teaching, having a PhD puts you into a very elite group which is competitive for the best teaching assignments, the best students, and so forth.  If you have a PhD, I think you would have an easier time bringing home a salary upwards of $75K as a high school teacher than as a non-tenure track college faculty member.  


  Would the PhD be enough to teach in a high school?


In many states, public high school teachers need to be certified.  My understanding is that certification is not require for private high schools (though this too probably varies by state) and I know of at least some private high schools which do not require it.  Anyway, though I have not done it, my understanding is that certification is a fairly easy process compared to getting a PhD.  In fact many students with teaching interests acquire this certification before they graduate, so it can't be that bad.


  And also, is it gonna be much harder for a non american to get hired at a high school rather than a college?


Sorry, I don't really know.  My guess is that if you do not already have a green card and are looking for immigration status coming from your teaching job, that will be harder.  If you are a permanent resident, speak fluent and intelligible English, and have a PhD (again, this confers highly elite status among high school teachers) from an American university, you're going to look like a catch.  Do recall that America is a nation of immigrants.  If you're teaching in an urban environment, there will be many other foreigners and first generation Americans milling about.  If you go to a small town, maybe they'll chase you out with pitchforks...but probably only if you have bolts sticking out of your neck.  


  Would a position in either of the 3 types of institution give me holidays for the summer? I assume high schools are closed in the summer, but maybe colleges will require to teach during summer.


Yes, you should get summers off in all three positions.  For high school this is of course one of the major perks of the job.  In a college or university they will probably offer you summer teaching, and they may pay you little enough to make that attractive, but you certainly don't have to.

Let me put in a vote for high school teaching.  America needs competent, intelligent, serious high school teachers more than anything else I can think of.  Like many jobs, it has some negative aspects, but it's truly important, good work.