Slightly off topic - but sending a dummy mail can be "dangerous".
Many servers "note" if you try and send to a local address, which does not exist. For example - if the server's domain is "whatever.com" and you send to a random address, say [email protected], and "aaa" is not a valid user, then the server notices this.
The server may then take an action like blocking you, as a sender, for a period of time. (This helps to reduce spam from dictionary attacks.) So your "test" ends up effectively blocking the real mail from being delivered.
The reverse is also true. Let's say you try to send to an external address, which you know is valid (your own email address for example) as the test. In this case the from address must be a valid internal address. If you use an invalid internal address, or worse an address which is not internal, it's likely the server will refuse to deliver the mail (at best) and at worst again institute a temporary block.
The key factor in both these situations is that although the SMTP protocol is very "loose", SMTP servers watch very closely for "bad behavior" because this is one way of distinguishing a spamming program. So any hide of "incorrect" behavior can lead to it arbitrarily refusing to accept your mails (usually for a limited period of time.)
Incidentally, back to your original question.
Both of your tests are pretty much instantaneous. Even if the email server is on the other side of the world you can do both checks inside a couple seconds. So chances are even if you send back 2 packets, to the user they'll appear as "arriving together". And since 1 request from the browser can only handle 1 response from the server you would need to send the response in 2 packets.
ie do first test - send first part of response - do second test - send second part of response.
For a normal HTTP packet this is no big deal. Do some sort of flush / send after the first response is ready, and then again after the second response. The browser is used to displaying partial pages as they arrive.
However for an AJAX request you'll need to get into your framework at quite a low level. Most frameworks, that I'm aware of, require the incoming Async packet to be "complete" before they start to parse it. This is especially true if the packet is formatted as say xml where partial parsing is useless in pretty much all cases.