1. This is the time when a woman can also be the cox.

2. This is the place where a woman can also be the cox.

3. This is the reason why a woman can also be the cox.

4. This is what I want to buy.
4-1. This is the thing which I want to buy.
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Q1: Are they all grammatical?
Q2: Is the cox a specific cox? Or is 'the cox' used in generic use?

The sentences are all fine.

"Cox" is the shortened form of "coxswain," and there's only one per boat, right? In your sentences, you're using it generically. If you wanted to use it specifically, I believe you'd need to name the boat or the team.

http://www.onelook.com/?w=cox&ls=a