Are there "NP-Intermediate-Complete" problems?
I don't really have references for these results--they aren't hard to prove once you understand Ladner's theorem.


No, for any NP-incomplete set A there is another set B strictly between A and SAT.
These equivalence classes are known as polynomial-many-one degrees. You can embed any finite poset into the degrees below NP. In particular the degrees are not totally ordered or finitely branching. 
This all depends on what you mean by "interesting". There is a huge theory of the degree structure of the computable sets (see Soare's book for instance) and many of those questions have not been ported down to polynomial-time sets. For instance, can you have NP sets A and B whose join is equivalent to SAT and whose meet is equivalent to the empty set?