Advice about job applications in the US (mathematics)
Answers to your questions (the numbering is the order you asked them):


You should definitely email people at the university who might be interested in and familiar with your work.  But for tenure track positions it is weird to say you want to work with them.  Instead, I would just send a low-key email just letting them know that you applied to their position and would be very interested in coming there.  The goal is to be informative but not pushy.
It is true that it is both more expensive and more difficult logistically to interview someone who is coming from overseas, but it certainly does happen.  They just have to want you enough.  For postdocs, there generally isn't an interview, so it isn't an issue.  However, some kinds of postdocs are funded by sources like the NSF that only want to support US citizens or permanent residents, so foreign applicants won't be considered for them.
It's not weird at all.  Doing so will not affect your chances at getting a tenure-track position.  Everyone knows that the job market is tight.  
There has to be someone at the university who advocates for hiring you, which means at least they appreciate the kinds of math you do.  But certainly there does not have to be someone who works in precisely the same part of mathematics (and in fact often the "needs of the department" involve hiring in fields that are not well-represented there).


My background: I've served on a large number of search committees in pure mathematics at a couple of different private research universities.