If you sign up for the FE upgrade program, there has to be someone to replace your FE seat in order for you to move to the Falcon. Unfortunately, there's nobody available ( or making it through training) so they hire dozens of street FOs for the Falcon before you ever get a chance, if you ever get a chance.
With regard to rest rules, if you say no, they will either fire you, or put you in "time out" where you don't get assigned flying for a long time.