I get a bad gateway error and can’t load the site and Cloudflare tells me it’s a host error (they are doing their job just fine, the first two bullet points assure me) so hopefully hosting a website isn’t what to do with it.
It depends on how the site is configured in Cloudflare. I'm pretty sure I had to change the default settings to make it keep my site up even if my server goes down (and even then I'm not sure I did it right)
The default settings don't cache HTML. (Maybe even if the server says to cache it. I seem to need to add special rules to make it respect the standard cache headers)