In the privilege page, the list of the privileges you see at the right is ordered based on the minimum reputation required for those privileges.
If you ask why in a beta phase you get first the privilege of voting to close a question than editing posts, that is because having users who can vote to close off-topic questions is more important than having users who can edit other users' posts.
Consider also that pro tem moderators are appointed during the public beta phase, but not at the beginning of that phase. If there would not be users who can vote to close questions, the new site would be filled with off-topic, or other bad questions in the moment that the site is defining which questions are allowed. If the reputation for voting to close questions would be higher, questions would be closed too late, and new users would see a question being asked, and would ask a similar question, with the result that, instead of having an off-topic question, you would have two off-topic questions.
The only alternative would be flagging the questions to close, but as there are no moderators for the new site, the only users who could close the question would be community managers/coordinators, who are not part of the community being built around that site. Stack Exchange wants the sites to be driven from the same community that uses that site.
So who decides the order? Is it done manually by moderators, or is it some automatic function based on some site metrics? Does a Stack's status (Beta or not) affect privilege order directly?
It's the status of the site that influences the reputation required for each privilege: The reputation is different if the site is a private beta, a public beta, or a graduated site.
Every site that in a public beta site will require the same reputation for the same privileges.