No. If your instance is down, you cannot use that link.
Correct, you need Discourse to be running. As I previously mentioned, you can download the backup by any other file transfer system you like. If Discourse isn’t working, using something like FTP or SCP would be suggested.
I’m fairly certain this is a one time link. However, Discourse is what tracks usage of the link, so if you click it while your site is completely down, the click should not be counted.