The following is my personal (i.e., includes all of my mathematical prejudices) ranked list of subject area journals in number theory.
From best to worst:
1) Algebra and Number Theory
2) International Journal of Number Theory
3) Journal de Theorie des Nombres de Bordeaux
4) Journal of Number Theory
5) Acta Arithmetica
6) Integers: The Journal of Combinatorial Number Theory
7) Journal of Integer Sequences
8) JP Journal of Algebra and Number Theory
For a slightly longer list, see
http://www.numbertheory.org/ntw/N6.html
but I don't have any personal experience with the journals listed there but not above.
Moreover, I think 1) is clearly the best (a very good journal), then 2)-5) are of roughly similar quality (all quite solid), then 6) and 7) have some nice papers and also some papers which I find not so interesting, novel and/or correct; I have not seen an interesting paper published in 8).
But I don't think that even 1) is as prestigious as the top subject journals in certain other areas, e.g. JDG or GAFA. There are some other excellent journals which, although not subject area journals, seem to be rather partial to number theory, e.g. Crelle, Math. Annalen, Compositio Math.
Finally, as far as analytic and combinatorial number theory goes, I think 4) and 5) should be reversed. (Were I an analytic number theorist, this would have caused me to rank 5) higher than 4) overall.)
In my experience, it depends on the specific funding of the program. Some programs funded by the NSF will require the applicants be Americans. But this is not always the case. Last year at my university, a student from Cambridge took part in the REU.
When applying for funding for these programs, the directors must include some explanation of who will be the participants. Additionally, many of these programs specifically encourage minorities to apply, sometimes including non-American applicants.
I encourage you to look at the specific programs to see if they allow foreign applicants, emailing them if it is not stated on the webpage.
Here is the webpage that I used when searching for an REU as an undergrad.
Additionally, just googling "NSF REU 2010" is helpful.
EDIT: Shameless plug: http://www.math.ksu.edu/reu/sumar/
Best Answer
The American Mathematical Monthly.
See also this list.