If you are receiving the "Cannot write in game folder" error, try creating a new Windows User account and see if you can install the game properly from that account.
Please note, however, that Windows® 10 is not officially supported by Dying Light, as the game had been created before its release and supports 64-bit Windows 7/8/8.1.
You can also try Re-installing the Game, making sure you have deleted any files related to Dying Light, such as C:\Users\USER_NAME\My Documents\DyingLight.
Having re-installed Dying Light, Run the game and Steam as the Administrator. In addition, try setting the game's executable to launch in Windows 7 compatibility mode.