Do my crime statistics update when I save the game or over a period of time? Everytime I quit the game and go look at my stats, they don't seem to have changed much.

You need to link your game to your account. Rerun the LCPDFR installer and say to link the game to your account.

Oh and just to further the post above, 0.91 can not link to the site because of API changes, so if you want your crime statistics to update, use 0.95 RC1.

Also it has nothing to do with saving game. I'm not sure why anyone would want to save a game of LCPDFR anyway as there is no progression but anyway the stats are updated normally each hour depending on server load. And as mentioned you need to link your account. And the post above me makes a valid point about version 0.91