Most web servers do out of the box, so I would assume most do. Basically all unless they have some reason to turn range processing off or are running a custom/experimental/both server that have implemented the feature (yet).

Not supporting range requests would be a disadvantage for any service hosting large files. Resuming failed long downloads wouldn't work so users might not be happy and there would be more load on your bandwidth and other resources as the AU falls back to performing a full download.

Generally yes. Because not having range support means you can't resume file downloads. Which is a pretty essential feature for a static file host.

More interestingly, do reverse-proxies like Varnish / CDNs like Cloudflare support range requests? If so, do they fetch the whole content on the back, and then allow arbitrary range requests within the cached content on the front?

Yes, Cloudflare behaves as you describe.

I was wondering that too. Support was spotty in general ~20 years ago but I assume things have improved since then.