Ads are listed in the order by which they are posted, so the newest appear at the top first. If someone pays to relist their ad, then this can move the listing to the top. Outside of that, featuring the ad is the way by which customers can pay to have their listing more visible on the homepage.
Yes, you can offer free ads with only payment required for featured listings if you wish. There is no featured sellers widget available at this time.
The advanced search functionality can be seen in the category view page as shown in the
ClassiPress demo here, on the right hand sidebar as "refine results". You can search locations and custom fields as determined by the site admin. Thanks :)