email as a countable noun?

As you can see, the people responding to your question don't all agree. It's a word that is in the middle of change. In the early days of email- then 1980s- native speakers of English mostly treated it as uncountable, parallel with "mail". But I started noticing that use in other countries, where the term was borrowed from English, tended to treat it as countable, and having a plural. I still use it as uncountable, but I'm an English teacher! Dictionary.com gives examples of both usages, but also has the sentence "Send me an email about that idea, and I'll get back to you."   The use of "an" in front indicates a countable noun. Eventually, common usage determines these issues.  

Not sure what you mean by "countable."

"I received three emails today." It's always been a countable noun. Not sure why you think it wasn't. Maybe because in post office terms "mail" is uncountable and "letter" is countable, but on the internet "email" has both meanings.

He sent me two emails asking where his package was.  Yes, it seems countable to me; been like that for years as far as I can remember.  It can also be used as a generic uncountable though: Sending email is a waste of time.

Apparently "email" appears countable, uncountable nouns are known to be measurable, not countable.

Yes.  I got ten emails today.