@dmoore, thanks a lot, that worked for me!
@oBFusCATed, yes, I want this to work on two monitors, but it's fine, I'm ok to have one same window, I just place the separation bar at the border line of two screens.
An interesting observation though: after performing the suggested manipulations with tabs the "Close tab" button (the X button) totally stopped working : ). Restarting the app didn't help, changing perspectives didn't help, I'm not sure how to get it back functioning. but fortunately Ctrl+W still works, so I can survive. Do you guys have any ideas why that happened and how to get it back to work ?