How can I have a Moderator star rating and the word "Moderator" display under the names of moderators like it does to my admin account?
Go to Forum under Admin Control Panel, then Moderator.
Select the user, the forum(s) and the permissions you want the moderator to have. Look at a forum where they have posted and you should see their rank updated.
Now for the other problem, you can try this.....
Still in the Forum Admin page, click on Forums, then try ReSync to see if your changes get updated.
Do you have
Extra User Settings: Enable optional extra user settings enabled?