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Cloudflare has a pretty extensive API. Otherwise, I'm sure the AWS APIs would cover this use case. But, with both you'd need to code something up, don't think they offer something like this out of the box.
I think so, too. We are using Cloudflare for SaaS to be able to cache requests to custom domains without owning them.
However, registering domains for our customers with them being in control and owning them, is another thing entirely :)