It seems like everyone does it. I've never heard of a paid moderator (though I'm sure they probably exist). From what I've gathered, however, it sounds like having any kind of volunteers is illegal for any for-profit business in the US, based on FLSA regulations. Is this true, or am I reading it wrong? Would this also apply to other types of volunteers (such as someone who wants to help code the website)?
An employee is someone that the employer "suffers or permits to work" - moderators would appear to be caught by this. There are specific exemptions carved out in the public and not-for-profit sectors where they "a) work toward public service, religious or humanitarian objectives; b) not expect or receive compensation for services; and c) not displace any genuine employees." However, they very specifically say “Under the FLSA, employees may not volunteer services to for-profit private sector employers.”
So on the face of it, a moderator is an employee and is entitled to minimum wages and conditions for the hours they work.
AOL settled a lawsuit in 2009 with their moderators who were suing for wages for an undisclosed sum and so the case did not set a precedent. This article suggests that "for-profit companies don’t have volunteers; they have lawsuits waiting to happen" and uses examines the situation at Reddit (which could equally apply here). Facebook employs moderators so the precedent exists that this is work that employees do.
When the lawsuit happens, we'll find out. It will turn on the particular facts - some types of mods for some companies may be employees while others may not.