There is no option on this yet but can be implemented through custom coding.
1) Does Classipress have the ability to accept payments keeping the user on site vs. having them leave the site with a redirect to PayPal to enter their information and then come back. If not is there a Plugin that can enable this?
2) Can you enable recurring monthly payments for the users who post? They can continue to post or have their post as long as the subscription is live with PayPal?
If the price is not required, there is an admin option to have it blank or display zeroes.
3) If you don't make the price required will it still display on the site?
No features on these yet and any feature request are to be posted at the ideas exchange page
http://ideas.appthemes.com/classipress/ so the users can vote for it.
4) Can you develop ad packages that have multiple functions enabled or disabled based on the package they purchase? For example:
$10 - 1 Image
$20 - 3 Images and featured listing
$30 - 5 Images, video and featured listing
$40 - 10 Images, video, featured listing and ad placement rotation
Sticky post is the same with featured post. Stickied posts are then displayed at the home page slider.
5) I read that you can create some posts as sticky posts and some as featured. Are these the same or can featured actually be listed below other normal posts?
No option on this yet but it can be done through custom coding. The theme is not encrypted and so it can be modified according to your requirements. Implementation is just dependent upon your coding skills or you can just hire a programmer to do it for you. Appthemes also have certified community partners
http://www.appthemes.com/partners/ that can be contacted for customization jobs.
6) Can you display ads through Google's Ad Serving Network (DFP) on the header ad position? The goal is to be able to have rotating ads on the site and not just one fixed ad.