I’m suspicious. If it were legit, I’d expect them to give you a link to logon, not ask you for password and birthdate info that they already have.
Why not search around the web?
Besides, if its Hotmail, if they kick you off, they’ll provide a way for you to secure your system and log back on in good graces. So, you don’t have to do anything until you have more info.
Wrong! Spammers can easily make a link look like it's legitimate and send you to their server which will look like a valid login page. NEVER click on a link from any email that asks you to confirm personal information.
When in doubt, login independently to the site in question-- in this case gmail, and go from there. NOT from the email.