Is the MinMax/optimization/search variant of a decision problem always easier/equal?
No.  A reduction only proves that it is "at least as hard", not that it is easier.  Normally, the optimization problem is approximately as hard as the corresponding decision version (to within a polynomial factor).

If the decision problem is in P (can be solved in polynomial time), and there's a polynomial time reduction from the optimization problem to the decision problem, then that implies that the optimization problem can be solved in polynomial time.  It's not literally in P, because P is a class of decision problems, but you can think of it as being in some class that is analogous/similar to P.

It's not clear to me what you mean by "the MinMax variant" or "the max-variant", so I can't answer that part.