While checking the rights, make sure you chose your active user and not the default one. Then, makes sure the active user has read, write, and execute permissions for the entire folder and its contents. This should cover the "rights" aspect of the problem.
No, shutting down the game by a cross icon instead of menu exiting shouldn't affect anything except if you did that during the saving process itself. Even then, the save file should be in the save folder.
As for the antivirus software, most of them have an exclusion or ignore list. Try adding the game's folder to that list and test it out by starting a new game and making a save once you'll have the control.