Do older repositories no longer work with newer Ubuntu version?

Is there no backwards compatibility between repositories?

Yes, there is but to some extent. For Ubuntu versions you can use repositories through the versions that are not end of life but mixing releases is not recommended.

1 problem you will run into is when there is a big change in the default software. Ubuntu went from Gnome to Unity so any tool depending on the old style will be buggy or completely useless. That probably includes that Mint Menu.

If someone stops maintaining a repository, is it basically dead and useless?

Ubuntu will move the default repositories to an archive so these are still usable to Ubuntu users. How others deal with this is up to them. Personal archives tend to be available on launchpad for all releases (even those end of life).