Is computing the dual optimum of a degenerate LP equivalent in complexity to solving LP?
Yes, in some sense at least.  See the following paper: 

Megiddo, Nimrod. "On finding primal-and dual-optimal bases." ORSA Journal on Computing 3.1 (1991): 63-65. PDF available here.

It shows that:


  if there exists a strongly polynomial time algorithm that finds a basis which is optimal for both the primal and
  the dual problems, given an optimal solution for one of the problems, then there exists a strongly polynomial algorithm for
  the general linear programming problem. 


(Note the details of the result though: it is about finding an optimal basis and deals with strongly polynomial-time algorithms.)