When do you think it's alright for a child to get an email address?

[Froggie]

Since Yahoo Answers allows participation at 13, that is a good guideline for age-appropriate web activity such as e-mail and Facebook. Do remember that anything you post online or send in e-mails can end up in web searches practically forever, and set up very secure passwords to prevent hacking.
Be cautious on the web - don't believe everything. http://www.snopes.com is a good source for checking out web scams and other Urban Legends. Never EVER open Spam mails in curiosity - the newest malware can infect a computer just from opening it!
You sound responsible enough for e-mail, and even a Messenger account. They are cheaper to use than cell phones, which most teens have anyway these days, even those smart phones with web capacity.
Tell your mother it is a reasonable activity, and appropriate for your age group.