What exactly happens at the financing dept?

That's assuming you're doing finance through the dealership. In my case I wasn't and the one and only thing the finance guy had to do was take my deposit. The paperwork was done by the time I got taken there.

This is also a fair point... I went in really late on a Saturday to a test drive and by the time we were doing the deal, people were going home. In fact the third person who sells the useless stuff had gone home and tried to chase me later via e-mail and phone (and got rid of her with a simple "Yep, read your e-mail, not interested, don't need it").