How would I send this email to the server along with the key?
You set the email for the product key when you generate the key (or even afterwards, if you want). That is, when a customer orders your software generate a key right then and there and set the email for the key.
Does that make sense?
Additionally, how would I make the license key dependent on this email address such that if the email is incorrect for the key, LimeLM will reject the registration attempt?
You can't do that. Why would you need that? A customer is prevented from sharing their key by the very nature of the design of hardware-locked licensing. So limiting a customer to using a key if there email matches is unnecessary.