If you are going to work a few weeks for free, you may as well think of a good small business application and build it. At least you will make money from it or you can add that to your resume.
The IP assignments I've signed when joining a company (and always gotten modified in my favor to varying degrees) have never said anything about whether or not they have their shit together enough to pay me. In my experience, they tend to boil down to: anything you do on company time, with company equipment, or at the company's request belongs to the company, full stop. You might be able to argue that you weren't on company time if you weren't paid for that time, but the other two clauses should cover that handily.
At my present job, I use my own laptop, and got the agreement winnowed down to just the third of those -- they only own my work if they specifically asked me to do it. (Granted, I'm not writing a lot software here, and what software we do write just exists to facilitate our core business, so I doubt they'd have any qualms about any of the developers modifying their agreements similarly, anyway.)
Right, but I seem to remember something in various state laws about a contract not being a contract unless there is "consideration" involved, i.e. both parties are receiving something of value. That's why so many transactions are for $1, eg. wealthy CEOs and Arnold Swarzenegger taking a salary of $1, old school buses or old surplus military equipment being sold for $1, etc. Regardless of what's written down, the transaction isn't a valid contract unless both parties receive something.
IANAL, this is all just idle speculation, consult with a real lawyer before you actually rely on this.
That's pretty much what I remember from my business law class, but then again I was often annoyed by the professor's little skits and rarely frequented whole live lectures (search Robert Emerson University of Florida if you haven't had a chance to see what nightmare the students go through).
