Bizarre Skype automatic logout issue
Try the old fix of running Skype once as Administrator, entering fully the credentials, then exiting and passing on to normal running.
This trick allows the program to update some parameter that it is stuck on trying to update, but cannot manage to do because of some snarl-up of permissions, and this may also reset those permissions.
This trick has worked for me, although for other products, and may also work for Skype.