If I'm selling a plane that has been de-registered, can the new owner re-register the aircraft with the same number (or do they have to get a new one)? Does the airworthiness certificate transfer as well? How long does this process typically take?
The FAA has a page that sort-of takes you through this process. Basically it boils down to this:
If your registration has expired you still have ~90 days to register the aircraft
So it's a "new registration", but you can keep your N number. (Yes, you have to use an official, FAA-Provided 8050-1 form - you can't download them anymore. You're also going to want to talk to someone at the Aircraft Registration Branch or your local FSDO before you start filling out forms because you don't want your form to show up after the 90 day window or you wind up in the other category...)
After about 90 days your N number is Cancelled & unavailable for 5 years
At this point you have to talk the aircraft registration branch, but my understanding is you are out of luck -- That N number is void and a new number will be issued when you register the aircraft (at which point you must apply the new N number to the aircraft in accordance with FAR 45, which typically means repainting at least part of the tail).
In both of these cases your airworthiness certificate is invalid* (because without a valid registration you have no authority to operate the aircraft).
In the first case you will not need a replacement airworthiness certificate (because your N number hasn't changed), but if your N number has been cancelled you will need a new airworthiness certificate issued with the new N number on it.
Total time for all the paperwork to be processed depends on which bucket you fall into -- the aircraft registration branch usually has about a 30-day backlog (check here), plus time for the post office to move the paper from point A to point B, so a lag of a little over 30 days is what you should expect.
The N-Number in question cannot be used for 5 years after it expires.
The airworthiness certificate does not change, it is just invalid without current registration. They may, however, send a new one upon re-registering if the registration expired, if the N-Number changes, or if you request a new certificate.
The process usually takes about a month.
you can register the aircraft in a different jurisdiction and use the same last part of the registration and change only the first 1 or 2 codes