With ClassiPress it is possible to have a mixture of free and paid ads with the use of the
Ads Packs functionality (or even with the "Price per category" pricing plan - but that of course is specific to categories). You can set the price and duration of these packages for distinction between the free and paid ads, but not the ability to charge only after a certain period of time.
If you were looking for a solution that didn't involve customization of the theme in any way, you could potentially make use of the membership packs functionality which could see you offering a limited time price for an unlimited amount of listings in a set time frame - which you could manually assign to users (although depending on the number of new registrations you expect, this could be more of a task to manage). Outside of this, it would require modification of the default theme to implement other payment options as you describe.