I recently submitted mine in Dec 08.
I started taking my classes in Nov 05. (Yes, it took me 3 years). Back then, the STAR Center issued a class certificate and each of individual control sheets, which made my package to the REC very thick. Now they just issue a certificate, and the completed control sheets are listed on the certificate.
To my knowledge, there are no required on board assessments. I certainly did not submit any, and though my application is not yet approved, I haven't been told any are necessary. It is my understanding that the celestial control sheets may be completed on board. However, I found the class a good refresher. Unless your celestial is very strong, I highly recommend the class.
One note that I wrote about on another thread: You must now submit your flashing light certificate before the application will be approved. I learned this the hard way. That is the hold-up on my approval.