nginx enables it by default. Another ingenious use of range request is zsync, it allows you to diff compressed binaries on a remote with local ones, so that you only have to download what has changed on an update. AppImage uses this
I can't think of a specific one, but a decent proportion (maybe a quarter?) of HTTP downloads I attempt don't support resuming a partial download, therefore don't support Range. (I.e. resuming always starts from the beginning.)
I would assume this is often because the site in question isn't using Apache etc. to serve a file directly, but is either essentially proxying it to some custom-built file serving service, or a script that processes/authenticates the file in some way, and they just never bothered to implement Range.
Depends on what’s being served. Any decent static file server should support it, but if the content is at all dynamically produced then the authors would have to think to implement it and rarely do.
But as long as you're dealing with a known server that does, then gravy!