Mods are selected based on multiple criteria such as general site activity and report success ratio. If we feel a member meets the criteria, then we may invite them to join the mod team. More information can be found in the FAQ link I provided. :)
Side note: Mods only recommend content to admins that they think violates the posting guidelines but admins make all final decisions so it would be pretty difficult for a mod to rule anyone with an "iron fist." Lol :P
You can certainly ask to become a mod. You do need to be up-to-date on the rules of the site, which you can read on the FAQ. You should also not have a ton of posts removed as that shows you wouldn't be able to moderate as well.
I think you can reach out to them, though the invitation is usually extended to you. You need to be active and provide the community with quality content. Read more here: http://www.girlsaskguys.com/faq/moderators