I’m not familiair with Synology, but in theory it’s perfectly possible to get a certificate for system X on a totally different system Y. For example, if the host to which the hostname in the DNS zonefile points isn’t available at all, you could always go for the
dns-01 challenge, assuming you are capable of adding TXT records to the DNS zone from system Y. You could even make the necessary DNS changes on your mobile phone, for example. You’d need a ACME client on system Y of course, which might be difficult on a mobile phone But assuming you have some kind of computer (since Certbot also works on Windows, it could even be a Windows PC), you can generate the certificate manually through the DNS challenge and manually move it to the Synology.
That latter part is also in theory, because I have no idea if Synology makes it possible to install certificates manually…
Yes, renewal is in essence just a new certificate with exactly the same hostnames as the previous certificate. Technically, it doesn’t matter if you “renew” after one day or after 365 days (not regarding the Rate Limits obviously ).