I just checked out a free demo earlier and I checked the 'stay logged in' box when I logged in. I went in the admin cp and after a few minutes,I would click on something or refresh the page and it would show the admin cp login. My suggestion is; when you check the 'stay logged in' option,it should also apply for the admin control panel.
I think you think I meant for it not to require a login to admin cp. I mean you should still log into the admincp after you logged into your admin account,just when you log into the admin cp,the 'stay logged in' option applies to the timeout of the admin cp so it won't time out when you have checked the option when logging into your account. See what I mean now?
Sessions can last quote a while. As you are using an IP, the session invalidates if the IP changes. For that to not be the case (as you suggestion with the stay logged in), it would cause security concerns.