Pure vs Applied maths, and combinatorics, in American universities
If you come here for "general advice", I am afraid you'll be disappointed. 


You should choose based on the people there. For each school you apply to, look through both the applied and pure departments and see where the mathematicians who work in combinatorics fit in. Most large research universities will list their faculty by research interest, or have dedicated research group webpages. For example, based on this page you probably do not want to apply to Princeton's pure math program. 

Note that many schools do not, at least on the graduate admission level, differentiate so strictly about going into applied versus pure mathematics. 

(For choosing the school, since you already have research experience, you should look up articles which you find interesting and find out where those authors currently reside.)
It really depends on the school. If a school has no combinatorialist in its pure maths department, and you apply there with an intention of studying combinatorics, your chance of admission is practically 0. Similarly in reverse if all the combinatorics is done in the pure maths department for the school you are applying to. 

There is no "one size fits all" solution. 
This again depends on the school. Some schools allow it, some schools don't. In many departments it is expected that you find an advisor from within the department and do your dissertation research on a subject that your advisor is interested in and/or an expert on. In other departments more leeway are given to students interested in more interdisciplinary subjects to be jointly advised by two advisors (possibly from different departments with different expertise). But remember, if you are going for a PhD you will need to find an advisor in any case. Shoehorning yourself into a situation where it maybe difficult to find a professor in your own department who is willing to advise you is, in my view, generally not recommended. 
This again depends on the individual policies for the schools. Some departments have very strict requirements on what the students must learn for their comprehensive exams; some, not so much. Most schools with graduate programs have very clearly written information on their websites about what is expected in their degree plans. For example, here's Harvard's version for the Applied Maths degree, though it seems they are a little bit short on the exact details. 


The only "general advice" I can give is this:


  Don't apply to graduate schools blindly, especially since you have a confirmed research interest. Do your homework and find the experts with whom you would like to study, and apply to study with them. Write to them in advance to confirm that they are interested in taking on students, and possibly solicit advice about other possible individuals if they are not. 


A good research school may not be strong in the field of research you want to do. Always, always check before applying.