Two batteries shouldn't matter.
No air intake or vacuum leaks doesn't bode well for your P0507 code. The next step is replacement of the ECM, according to the FSM. You may end up needing to go the dealer to fix that code.
The possible cause list for P1456 is a lot lengthier (
see page EC-645). I'd check all the EVAP vacuum hoses around the charcoal canister to make sure they're not cracked or loose.