Steven Zahm@ Mary
RE: is there a payment extension for new members signing up to join our association and is there a way to have them pay their renewals each year.
Sorry, no. But in my opinion this is a good thing. I recommend you use a membership plugin like Restrict Content Pro to manage membership payments and renewals. This greatly increases your ability to monetize your site where a director is just one benefit. There are other directory plugins which do integrate payments, but then you lock yourself into only being able to monetize your directory making it very difficult to expand member benefits.
RE: if a member wanted to update their information would they have to file a request for the admin to handle or can they update their own information.
You could manage that way it you wish. Alternative, you could use the Link extension. This would allow you to allow the user to maintain their own entry with or without moderation.
RE: is there a way to select a business to be featured on the homepage and can this business be easily changed each month?
Actually I have nothing available at the moment to do this, sorry.
I hope I have answered your questions thoroughly!