1. I'm guessing just good old fashion detective work. He has an army of goons and plenty of resources. It's not impossible to figure out Batman's identity. Hugo Strange did it in Arkham City. 2. Assuming the Batcave is under Wayne manor I'm guessing Bane just stumbled upon it when he raided Wayne Manor. 3. This is harder to answer.
In Origins Bane is using venom for the first fight. For the second fight, Bane uses a precursor to the Titan formula, which results in his memory loss. Arkham City he is back on Venom but he is trying to get the Titan back because he is suffering from withdrawal. Batman stops him and in Arkham Knight, cash says he is probably out there coping
Even after everything you've done I would have saved you." - (Batman to Joker before the latter's death) "There's still work to do here in Arkham City." (Batman before being switched to Catwoman) "Go and hide. Now. (Prisoner or Doctor Safe while some with thugs as far) "This is not going to be pretty."
RS1ko.