Macbook A1342 2010 - OS Installation Issue
If you have access to the installer have you tried creating the bootable thumb drive from here?

Reading closer you don't have access to the installer.  Running the online installer won't work because apple pulled the download links that it uses.  I suspect this is thinly veiled attempt at pushing people to upgrade.

Check this out:  https://diskmakerx.com. It will create the thumb drive for you.  You will most likely have to go here and buy El Capitan from apple if you haven't installed already.

Its also worth checking out http://dosdude1.com/highsierra/ his tool will download the High Sierra installer for you from apple.  This is the last supported release for your MacBook.  Once you have the installer you can use DiskMakerX.