1. There are managers that will store them on their servers and others that are local.
2. You can sync it through something like Google drive/Nextcloud.
3. You should back up your password vault.
4. Your device may be compromised, but your vault is still encrypted. Really depends on what kind of hack it is.
5. You don’t really unless they’re an open source one like Bitwarden.
6. Yes. Instead of remembering a lot of passwords, you remember the master password to your vault
7. No. Because randomly generated passwords gated behind one secure password you remember is better than reusing the same/variations of one password.

You can try Bitwarden if you want a hosted solution that’s easy to use. Or, use KeePassXC and compatible mobile apps while syncing it through a cloud service. I do the latter.