Hello MILO,
The bug originally reported in this thread had been fixed back in march. If you’re still having some download happening, it’s probably caused by some data corruption or permission issue.
You should reinstall the Battle.net app (please make sure to have it installed in its default install folder on the C: drive if that’s not already the case).
For the game, please also delete and reinstall it, in a different folder and/or drive if applicable.
If it persists, we’ll need your MSinfo. You can have the informations hosted on an external site such as Pastebin then provide the URL here on the forums (in a new thread please), in a preformatted text. Else, you can just submit a ticket with the file attached.
For the patch happening after closing the game, we’re looking into it but it’s not doing any actual download after some tests for the moment. For new reports, please start a new thread.