After installing the DC3, run X-Plane but load one of the default aircraft, like the Cessna C172. You must get a window to enter your X-Aviation login details. This will activate the DC3. The load the DC3 and should work.
You Log file shows that has not been activated.
0:00:56.104 G64: debug: GK: Load license from disk..
0:00:56.104 G64: debug: GK: 404: license database
0:00:56.104 G64: debug: GK: Empty license file. Defaulting to login page.
0:00:57.097 G64: debug: No license for: (Leading Edge Simulations)(Douglas DC-3 v2 Published by X-Aviation)
0:02:44.229 G64: debug: GK: Requesting license update..
0:02:44.229 G64: debug: GK: update_license()
0:02:44.229 G64: debug: GK: No valid auth token, manual login required.
Currently there is a bug in DC3 that crashes if you load the aircraft when has not been activated. Fixed for the next update.