You should notice that what ever temporary alpha (angle of attack) or beta (angle of sideslip) is it has directly nothing to do with the flight path of the plane. Usually you use alpha and beta to control the amount of lift and side force the plane creates at any given moment. And after that when the force is created the plane might start to follow the given control and for example reduce these angles. So to say that alpha and beta are the steering angles but what the actual flight path will be is a different question.
What exactly is the airplane's stability axis?
It is the body axis when the angle of attack A.O.A or α (alpha) is zero. And you should not include the side slip β (beta) to that transformation. Here is an old picture which clarifies the definition.
"The body axis system is a right-handed set of mutually perpendicular axes whose origin is fixed with respect to the aircraft at the (35% M.A.C) point in the plane of symmetry. Aerodynamic forces and moments are given in terms of this fixed point. The x-stability axis is the projection of the velocity vector of the aircraft on the plane of symmetry. The angle between the velocity vector and the x-stability axis is defined as the slide slip β (beta). The angle between the x-stability axis and the x-body axis is defined as the angle of attack α (alpha). The z-stability axis is in the plane of symmetry of the aircraft.
The aircrafts manufacturer's aerodynamic data is usually presented in the stability axis system. This data is in the form of non-dimensional aerodynamic coefficients."
If you include the side slip in the system then it is the wind axis system. All three axes are shown in the following picture. The wind axis is also called the flight-path axis.
Wind (w), Stability (s) and Body (b) axis systems
Stability and Control Derivatives
There is a flight dynamics text book available from TU Delft, Delft, Nederland in the following link. It gives you methods to solve stability and control derivatives.
http://www.aerostudents.com/files/flightDynamics/flightDynamicsFullVersion.pdf
There is an Excel spreadsheet available that suppose to do the same from Cal Poly Flight Simulation Group, Aerospace Engineering Department, California Polytechnic State University, USA in the following links:
"Computes static and dynamic stability and control characteristics of aircraft based on geometric input. Implements DATCOM methods adapted from the Air Force as well as compiled solutions by Roskam."
http://aerosim.calpoly.edu/media/cms_page_media/15/Brad%20Beachler%20-%20SAD%20File%20Creator.pdf
http://aerosim.calpoly.edu/media/cms_page_media/15/Brad%20Beachler%20-%20SAD%20File%20Creator.xls
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