Not sure if there´s any software that sticks a tentacle out int the ether to bring all your E-mail accounts fully logged in and ready to download , Komp. Though Gmail is SMTP, I do know but swings both ways where you can either go out ont the net and read your mail or have it come home to you.
What you could do is group all your e-mail accounts as "home page" with Mozilla, for example; - ask Mozilla to remember all your passwords and whenever you want to check your e-mail just press the home page symbol.
Quite clever really, what.
bviously, if you already have pages open that you don´t want to lose you just open a second copy of the browser.