In episode 10.03 "The Dependence Transcendence" Colonel Richard Williams says: "You think you’re the first government contractor that wasn’t going to deliver on time?"

So I think they are getting paid by the government. I don't know what is in their contract, if they are getting a salary, or if they will be paid when they deliver the miniaturized gyroscope.

With the 2 months deadline and little sleep, working in the weekends and skipping a university party, it seems they were working in their free time. It's not clear if they took any time off from their university jobs.

Now that the deadline has been extended to 2 years they can go to the movies and eat popcorn.

The university is getting most of the money from the sale of the product. I don't remember which episode it was that they explained all of that. But I assume that means they are still working for the University when they are working on this project.

@Taylorfirst1 said:

The university is getting most of the money from the sale of the product. I don't remember which episode it was that they explained all of that. But I assume that means they are still working for the University when they are working on this project.

You are right. In season 9 episode 18 "The Application Deterioration" they go to the University Legal Office to file a patent for their invention. The attorney tells them: "... the university will own 75% of the patent." Anytime they use this patent in a project the university collects. In this case they are working for the government and through partial ownership of the patent the university will get the larger share of the profit. This would mean that, indirectly, they are also working for the university.

The episode does undermine Sheldon's assertion that he is a legal expert. Being a stickler for facts and accuracy he should have known all this beforehand.