I hit the 1000 reviews mark several days ago and still don't have the Reviewer badge.
(I saw that two of the new mods already have the badge, but they earned it before being elected.)
Yes, Moderators can earn the Reviewer badge.
I have it over on Web Apps where I have been a moderator since Feb 2011.
A post becomes "reviewed" if you click "review post" or the hyperlink from the /review page. You can act on it anywhere after that.