1. Can the Battleship participate in bombarding both territories? (ie is bombardment allowed once per COMBAT or once per ENTIRE TURN?)
2. Are there differences in rules between 2nd ed revised and Larry Harris Tourney Rules?
No, not when dealing with battleship bombardment.
3. Lets say that Battleship already in seazone that you want to assault from, if it participates in bombardment, can it also move during the non-combat phase? (I suspect no cause it was part of combat, but an opponent I played against insisted it could arguing it was simply a property of battleships)
No. Once it bombards its move is done.
4. If answer to 4 is "no it cant make a non-combat move", and you intend to move battleship during non-combat move, can you elect NOT to bombard (ie is bombardment optional or a compulsory action?)
Yes. You do not have to participate in the bombardment and if you don't you are free to move the BB.