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Yes you can sleep in the mind flayer region (though keep in mind that if you are doing this as the matron mother's quest to get the elder orb brain you have only three days (I think) to do it, so don't sleep too often).
To get out there are a number of things you need to do. You should have spoken to the githyanki who explained about the doors and needing to kill the elder brain to open the exit doors. Most doors in here are only openable by changing into the slayer or by a mind flayer, which you can do by using the control circlets (obtained in the slaves area, bottom right). Then you just need to explore around until you find the elder brain and kill it, which will open all the doors.
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