(perfect fractional) matching in regular hypergraphs
You are asking many things in the question, and I can only answer part of it:


  necessary and/or sufficient conditions for existence perfect matchings


It is unlikely that a nice necessary and sufficient characterization is known, because deciding whether a given 3-regular hypergraph has a perfect matching or not is NP-complete.  Indeed, it is equivalent to the monotone 1-in-3 SAT problem, which is proved to be NP-complete by Schaefer [Sch78].  (In [Sch78], the same problem is called “ONE-IN-THREE SATISFIABILITY” without the word “monotone.”)  To see this equivalence, interpret each hyperedge as a variable and each vertex as a clause.

[Sch78] Thomas J. Schaefer.  The complexity of satisfiability problems.  In Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC), pp. 216–226, May 1978.  DOI: 10.1145/800133.804350.