Moderators have very little power, are not paid, and are not something you should be worried about. If anything, you have just as much "power" as they do. If you find a moderator abusing their duties, simply contact the administrators. But all in all, they are no different than regular users, albeit a few authority options, so unfortunately they are just as able to become out of control as we are.
Regarding age - only admin (i.e, crown next to user name, like mine) can change the age, and we only do that when a user states his/her actual age in a question or answer, or when we are contacted.
Mods are permitted to remove answers, and admins do review mod removals. Mods are chosen because they have demonstrated that they have a genuine interest in keeping the GirlsAskGuys community safe, helpful, and enjoyable.
If you would like further details as to why any posting was removed, please forward the email that we sent to you, and we will provide you with specific information regarding the decision to remove the posting.
Moderators can delete answers and comments, but they can't delete questions, or even change your age. Only admins can do that. As for abuse of power, well yeah, it happens. However, I think admins are notified when an answer or comment is removed, and I would assume they remove moderator status from people who consistently remove things for no good reason. I've yet to see that happen though.
Moderators are moderated themselves. Admins can see what we report and how much we report. If they believe that what we've hidden is perfectly fine, it'll be visible again. If a moderator keeps hiding things that shouldn't be hidden or are clearly hiding things they don't agree with, they'll be demoted.
If you ask to be a mod, I believe they check your profile, your grammar, any reports on you and how you interact on the site. If it isn't satisfactory, you don't get to become one. I'm a moderator now because I was offered a position due to regular activity, reports, ideas and being a good GAGer. It's not too hard to become one but it's pretty easy to lose your position.
I don't think mods can really do much of anything short of getting rid of answers are deemed mean, hurtful, insulting and generally not nice . They can't even see who posts anon comments. Admins can do pretty much anything, though.