The theme should be able to accomplish the following:
1. Visitors to the landing page sign up or complete a couple of questions
2. Upon signing up, the page generates a special URL for them to share with their friends (to create virality).
3. Use the URL to track your most active referrers and reward them for spreading the word.
The closest I have come to is http://launcheffectapp.com/ but their designs look terrible for our purposes. If I could replace theirs with a beautiful landing page that might work. Could anybody help?
You might want to try unbounce.com they have a free version that has their branding on it but allows for the use of most features and if its for a course then when they are done they can upgrade if they wanted to get rid of the branding. Just a thought take a look. Might be easier to setup then WP for some students,and built more specifically for landing page conversions.
You can use any theme for this purpose. and for the functionality you have to build a WP plugin that will offer the features.
I build a similar plugin a year ago for one of my clients.
It may cost 1K USD to build the plugin.