How important is the country of academic experience in applying for US academic jobs?
To answer your questions:


They do not require, but they will usually prefer someone who is familiar with the academic system that they use. For example, small liberal arts colleges will usually prefer people with experience with small liberal arts colleges.
Perhaps not "not competitive" but certainly "less competitive" unless you are in a Tier 1 school in a nation that is known to have competitive tertiary education.


People have already noted in comments that there are two types of deans at American academic institutions:


Academic deans who are heads of units (the Dean of the Graduate School, for example).  These are usually promoted from within the current faculty body but on rare occasions are brought in from outside.
Non-academic deans or deans of student services (the Dean of Career Services, for example). These can be promoted (or laterally transferred) from the faculty body but are more often hired for that particular role and specialty. Non-academic deans are much more mobile -- for better or for worse. They move from institution to institution, hopefully moving up the ranks.


I would say that in your case the possibility of moving into an academic dean slot are very small.  You do have much more hope for a position as a non-academic dean. 

I should note that one option that is open to you is to apply for a senior faculty position in a university with the intent of becoming Chair, then a Dean within that institution. There are often job postings listed for departments that explicitly want senior faculty -- including senior faculty who are willing to immediate Chair the department.