I think what you are asking is on the edge of what is doable.
To specifically answer your questions;
You should probably use the ROS package RGBD Slam made for this exact process.
Yes, it uses SLAM.
No, an arduino is probably not sufficient, or appropriate. It will take a fairly powerful PC class computer to process 3D SLAM data, and that would be independent of any Autonomous processing - so it might actually need 2 full function PCs before it flies. Plus an arduino for flight control.
Being autonomous is an open concept. You have to write software that does something. SLAM will allow you to estimate where you are, and what your environment looks like, but you have to program the instructions to do anything within that environment. There is no general purpose "let's be autonomous" program.
The Kinect is part of a toy. It is reliable enough for experiments, you should at no time use a Kinect in any place where there is potential danger to life or property, such as on an unrestricted 'copter'.
The 'copter' will have to either have remote processing, which would involve some fairly high bandwidth transmission equipment, or be able to lift 2 decent size computers, plus a kinect in either case. The rule of thumb for 'copters' is 2 to 1 thrust to weight ratio, so you are looking at a fairly robust platform with eight motors (typical), costing several thousand dollars. I'm not sure a $10,000 cinestar would even be up to it, more and likely something custom beyond that.
Just as reference as to whether or not it is doable, I found a video showing it (a year ago, and we've gotten a little better since then).