Hm, that's not a simple question to answer.
It's not actually the matter of importance but rather of which e-mails can wait. There is a client, a global corp. interested in becoming partners with us who's willing to add our software to their products, big bucks involved, and no matter what I'm doing at the moment I will answer their e-mails right away when they come in.
On the other hand I have a small client who's not in a rush and I know I can wait with him till Monday.
The bad habit is using the inbox as your to-do list, but that's what has actually become. Certainly I should answer all the e-mails promptly, as waiting for answer is really frustrating from the client's point of view.
Internal communication is most often the least important like what's for lunch or who's going to play the ball after work, and most of this stuff should be completely separated from e-mails sent by clients.
I think there also should be a text field for notes next to the e-mail's title so that I could write short notes on how to process the e-mail once I have time.
When I know I won't have time to answer a particular e-mail later I try to answer it instantly, but later I might forget to follow up. If I don't have time it's also good to know to whom I could forward the e-mail and know that the person will 100% answer it on the very same day. Nothing is worse than telling a Client someone else will fix the problem and then a week later you learn that your colleague actually didn't get in touch with them.