How to prove the optimization version problem (whose decision version is NP-complete) can be solved in poly-time iff P=NP?
If the decision version can be solved in poly-time, can the optimization version be solved in poly-time as well?


This depends on how technically precise you want to be.

The most correct answer is "No". Consider an optimisation problem $OP$ for which the encoded length of the output is superpolynomial in the encoded length of the input. Clearly no algorithm can produce the output in time polynomial in the encoded length of the input, so $OP \not\in \mathcal{P}$. However, the decision version $DP$ takes as input both the input of $OP$ and the guessed answer, so it is allowed time polynomial in the sum of the two, and in particular does not need to execute in time polynomial in the length of the problem per se.

However, if you're guaranteed that the output of $OP$ has (asymtoptically) an encoded length which is polynomial in the encoded length of the input (say $O(n^a)$ for input length $n$ and unknown $a$), then you can do a binary search, starting at any reasonable guess, doubling unless you get high enough, and then searching back down. The number of calls of the decision problem is $O(n^a)$, each call of the decision problem takes time polynomial in $n$, and the product of two polynomials is a polynomial.