I am currently logged in of course, but if some reason I get logged out I'll be in trouble since I don't remember my password. Is there any way to view my password and any other relevant account data while I'm logged in?
Not your password, no. Stack Exchange hashes your password before it's stored, and there's no way for them to retrieve it, all they can do it compare the result when they apply the same procedure to the password you enter when you log in. (If you didn't know this, I encourage you to read more about the topic; it's a practical application of theoretical mathematics.)
Some other data, e.g. your website, Twitter and email address, can be displayed and changed. Visit this link: https://math.stackexchange.com/users/edit/current (for email, you'll need the 'Edit email settings' tab on the left).
Here is what to do. It is the same as many other sites.
First, before you log out, make sure the email associated to your account is the one you want. (Otherwise, change it as Glorfindel explained.)
Next, log out. Then choose "log in". There is an option "forgot my password" to click. That will email a link to you using the email address associated with the account.