If I check the box that I want Firefox, etc. to REMEMBER my login/password info,
I think you might be confusing passwords stored by Firefox with those stored by websites in the form of cookies.
You can see which logins Firefox is storing in its password manager by going to Edit -> Preferences -> Security -> Saved_Passwords. Logins listed here should be remembered as long as you keep your firefox profile.
Some web pages are set up so that Firefox won't save the password. If these sites give you an option to save the password (often in the form of a checkbox) then a cookie is being used to store the login information. The expiry of these cookies is set by the website and can't easily be changed by the user.