you do not have permission to write to the location you have selected

Install Sierra or El Capitan from Scratch

1. Restart the computer. Immediately after the chime hold down the CommandandRkeys until the Apple logo appears. When the Utility Menu appears:
2. Select Disk Utility from the Utility Menu and click on Continue button.
3. When Disk Utility loads select the volume (indented entry, usually Macintosh HD) from the Device list.
4. Click on the Erase icon in Disk Utility's main window. A panel will drop down.
5. Set the Format type to Mac OS Extended (Journaled.)
6. Click on the Apply button, then wait for the Done button to activate and click on it.
7. Quit Disk Utility and return to the Utility Menu.
8. Select Reinstall OS X and click on the Continue button.

Yes. If you can backup your files in some way then I would do so. Your drive must have a valid Recovery HD. If you don't wish to do this on your computer's drive, then select a connected drive instead such as an external drive.

Do you have a copy of the Sierra or El Capitan full installer app?

Why would you tell someone to completely wipe their laptop and reinstall the OS when they just aren't able to write to an external drive???