How does the location of a propeller affect an aircraft's stability?
There is a misconception that putting the prop at the back is destabilizing in yaw/pitch, like you were pushing a car as opposed to pulling it.  That only applies if the thrust element has a flexible connection to the body it's pushing/pulling.
When rigidly connected, the thrust axis is the same at any location along the plane's length and has no effect on stability.  All that matters is how far the thrust axis is from the center of mass where it passes by.  If the thrust axis is above the center of mass, power will make it pitch down whether the prop is forward or aft, and vice versa for a low thrust line.
Beyond that, the location of the engine/prop can have inertial effects related to the mass of the engine relative to center of mass, and gyroscopic propeller effects that vary because of the different location of the precession source, and the tail pusher can actually have enhanced stability due to various aerodynamic factors like an effective fin area increase due to the extra surface area of the engine/prop aft, but the actual placement of the thrust element on the longitudinal axis is not part of it.
Engine/propellers are usually on the front because it resolves so many design compromises much more readily.  Pushers are tail heavy when unloaded, which is a big problem (tail pusher jets have to have their main gears farther aft than optimal for takeoff and landing so they don't tip back when unloaded, which requires a larger horizontal tail to provide downforce for rotation).  The prop is subjected a lot more to junk thrown up by the wheels.  Aluminum propeller blades can suffer from corrosion from carbon in the exhaust. All sorts of little problems present themselves, that go away if the engine is at the front.