For the mx, it's easy, as it's publicly available.
$>host -t mx sitepoint.com
sitepoint.com mail is handled by 10 mail.sitepoint.com.
On the other hand, for the C and A records, you have to know them.
You cannot ask for a list of them, only check if a given subdomain has been declared.
But I don't get it...
Don't you have a mean to see what the zone definition is, on the host server ?
Usually, the customer is able to make the changes he want via an management console.
If you don't have such a control panel, then you are out of luck.
As far as I know, you could have your server declared as a slave for the given zone, but this has to be configured on the master server.
And a zone transfert can only be done once this setup has been done.
Couldn't you simply put a new zone, with the known subdomains, mx and new dns and add a catch-all that would redirect to the main subdomain (like www) ?
If you cannot ask the current hoster for help, then there is little room for improvements.