I'm pretty new to this stuff to but why not use a solenoid (or something to that effect) instead of a servo/motor/cd-drive, i simple pull motion when power is applied then would only need to be connected via a hinge (not even for a sliding door) and not have a need for gears/pulleys or any of that stuff. I'd be looking at diminishing the time taken and noise created for the door to open so as to not deter the cats from using it (I have a jumpy cat that will not use a normal cat door), the cd-drive could be a little slow to open the door as it takes a fair bit time just to open the tray which only moves a few inches. You also then only have a plastic gear which you could buy at Jaycar for next to nothing.
a solenoid is similar to a dc motor but has the electro-magnet at the rear not wound around the stack housing the shaft, when power is applied the electro-magnet pulls the shaft back in to the housing, when power is cut springs pull the shaft out. These can be bought from stores such as Jaycar, a push/pull solenoid was used in the "putting partner" devices (golf training device) for returning the ball.
I would speak to the people at your local electronics parts shop as to which one would be most suitable however I would be looking for one that has a decent range of movement so you do not have to attach it to close to the hinge on the door, the closer to the hinge you get the greater the power required to pull it open. Other than that you want to make sure it works so that when power is applied the shaft is drawn back and therefore only requiring power to open the door not to keep it closed.as far as attaching it to the cat door, if it is a hinged door (flap) you may need one either side mounted to the actual (human sized) door. the shaft then attaches to the cat door using a hinge (elbow joint). You could use the dc power adapter on the Arduino and 5 volt rail to power the solenoid, using capacitors or an extra power feed if required. unsure on the operating voltage although I am sure there is a range to choose from.Cost would vary depending on the solenoid, its uses and where you are in the world. In Australia it would cost between $10 and $30 dollars for standard application solenoid, more for industrial type.BTW jury rigging a cd-drives eject button would only be a matter of removing it and attaching the contacts to a pin on your Arduino and sending either high or low signal.I do not suspect your cats will press a button in which case you need a pressure switch, light sensor or infrared beam in place of the button. A light sensor would need to be mounted very specifically so as to not open the door when the sun goes down. Not sure but from what I learned yesterday about floating pins this may be used to sense if a cat has walked between 2 probes but how to do that, if it would work on that scale or if it would be reliable I am unsure.