Just be careful.

I am not sure about the type of your position. Many CA postdocs, staffs,in non-profit organizations,including universities and national labs cannot get H-1B visa. Instead J-1 is the option. I knew somebody had really bad experiences with J-1 visa. A lot of time is wasted dealing with the waiver.

This is a post from mitbbs. I think it is true and useful.

http://www.mitbbs.com/mitbbs_article_t.php?board=JobHunting&gid=10507287&ftype=0

If the position is faculty, normally you willnot have this kind of problem.

To answer your questions, there ARE some universities that donot provide H-1B visa. It depends on your position.

If they do provide you H-1B visa, there is no cap limitation of course.