Glad the reinstall helped. It looks like it was probably a permissions issue, with the game not being able to write data to the folder the saves etc were in. That'd explain both the empty log and the error saving.
It was install into C:/.../Starsector.65.1
Everything, what is install on C:/ need permisson/ Adminright. ( Yea, a hell, when i start my computer...) It starts only with Adminrights.
Or, what is more possible:
I have delete the save folder, start the game, create new character, have this. then, re-create the save-folder, and start again, create again new character, have this. (But this don't tell me why the log is empty...)
Now, it's install on another device, have no problem with it.