"NP-complete" optimization problems
An attempt on a partial answer:

Decision problems were already investigated for some time before optimization problems came into view, in the sense as they are treated from the approximation algorithms perspective.

You have to be careful when carrying over the concepts from decision problems. It can be done and a precise notion of NP-completeness for optimization problems can be given. Look at this answer. It is of course different from the NP-completeness for decision problems, but it is based on the sames ideas (reductions).

If you are faced with an optimization problem that doesn’t allow a verification with a feasible solution, then there is not much you can do. That is why one usually assumes that:


We can verify efficiently if the input is actually a valid instance of our optimization problem.
The size of the feasible solutions is bounded polynomially by the size of the inputs.
We can verify efficiently if a solution is a feasible solution of the input.
The value of a solution can be determined efficiently.


Otherwise, there is not much we can hope to achieve.

The complexity class $\mathrm{NP}$ only contains decisions problems per definition. So there aren’t any optimizations problems in it. And the Verifier-based definition of  $\mathrm{NP}$ you mention is specific to $\mathrm{NP}$. I haven’t encountered it with optimization problems.

If you want to verify that a solution is not just feasible, but also optimal, I would say that this is as hard as solving the original optimization problem because, in order to refute a given feasible and possibly optimal solution as non-optimal, you have to give a better solution, which might require you to find the true optimal solution. 

But that doesn’t mean that the optimization problem is harder. See this answer, which depends of course on the precise definitions.