"Relatives" of the shortest path problem
Here is an answer to the first problem:


  Path with minimum weight gap: find an $s-t$ path, such that the difference between the largest  and smallest edge weights on the path is minimum.


A paper from 1984 shows that whenever we can decide 
feasibility (does there exist a solution in the unweighted
case) for some combinatorial optimization problem in 
polynomial time, then we can also find in polynomial 
time a solution that minimizes the differenc between the 
largest and smallest cost coefficient (in the weighted case):


  S. Martello, W.R. Pulleyblank, P. Toth, D. de Werra
  Balanced optimization problems
  Operations Research Letters 3, 1984, Pages 275-278


This implies a polynomial time algorithm to your question.