You could copy the relevant key configuration files in your "Save" folder and rename them to match your new character's name. In my case the file name would be "keys_lawrence.xml". I've checked with our technical staff if this could cause any issues. While they do not see any problems with doing this, please make a backup of your original file just in case.
I'll also see with the developers if we can provide an option to automatically use the same key configuration for all of your characters by default. Of course no promises, but it will be passed on with my next report.
As for logging out.. Your configuration will automatically be saved when you log out from the game normally.