Simplification of weighted NFA
By simplification I understand either minimization or determinization. I'll try to sum up what I know about both problems, in the quite general setting of weighted automata over arbitrary semiring. The original works were done by Marcel-Paul Schützenberger (who introduced them), and you'll find a nice account of what is known about them in the book Elements of Automata Theory by Jacques Sakarovitch (also available in French): 
For shorter explanations, check out the lecture notes by Jacques Sakarovitch again: 

For both minimization and determinization (called "sequentialization") there are nice theoretical answers. For instance, every weighted automaton over a field can be minimized in polynomial time (see for instance this lecture note).