I won't say you cannot or shouldn't backup a container since it it wrong imo, but as your using docker as a base i'd say just backup mapped volumes and configuration files. Docker is built around breaking and rebuilding your container and keep working anyways as long as your mapped volumes and docker configurations are the same (asuming you use a well maintained image for updates or something of your own).

That's probably the sane way to do it and will save you space. I don't see the benefit of a full container backup (unless it's an lxc, lxd container or another full persistent container type).

As for the backup repository you can use whatever. If it were me i'd use another server or a NAS, but most cloud storage provider would fit. Storage is never free though.