You can't use your own domain/certificate. If you CNAME it the cert will be invalid.
It sounds like you can get a stable URL for a function, but it'll still be a Lambda URL.
I'd love to use this with a custom domain, without having to use an API Gateway.
Yes, but you'll have to either set up a proxy or do a client-side redirect.
You can always proxy it (e.g. Cloudflare).
For a lot of use cases if you proxy it through something external like Cloudflare, you might as well just write the code in Cloudflare Functions. Even faster as no proxying will be required.
You can't run a docker images on Cloudflare functions, nor can you run a code with 10GB size.