Single motor driving multiple targets - possible?
Let me just state the obvious:  Use two motors.  Now hear me out on this one!

There is a saying that "If all you have is a hammer, then everything looks like a nail".  Meaning, people tend to view their problems based on the tools or abilities that they have.  If they have a hammer, then they will hit their problems into submission.  This approach can be both good and bad.

It's good in the sense that you will often find a solution that you have the ability to accomplish.

It is bad in the sense that you asked a mechanical question in an electrical engineering site.  You are going to get answers that prefer electrical engineering solutions.

That being said:  Gears are hard.  Building mechanical systems based on gears is hard.  Building one-off mechanical systems in your basement is super hard.  The reason for this is that mechanical systems are very "tweaky".  You have to factor in things like the flexing of material, the tolerances of your fabrication method, the ability to get parts in the sizes that you need and the price that is reasonable.  Doing any of this with gears brings the level of difficulty higher.  I wouldn't even consider making something like this without a 3-D printer, CNC machine, or laser cutter-- especially if you want it to be small and elegant/unobtrusive.

Even if you can build it, will it work on your first try?  Tenth try?  Is it going to be reliable?  Durable?  Is it going to be noisy?  Electronics are not so hard, especially if you are just driving a simple motor(s).  

Which is more difficult:  getting, mounting, and controlling a second motor or making a complex mechanical thing with gears and stuff? 

If I were building this, I would use two motors.  According to FedEx, my 3-D printer should arrive here tomorrow and I have been studying gears and stuff for the past month-- and even then I would still use two motors.   But then again, I'm an electrical engineer.