It really all depends on if you want to use the room or not. Hotels usually can store luggage for you. If you want to take a shower or change clothes as soon as you reach the hotel I would recommend to include the 12th and reach-out to the hotel mentioning you will be arriving late.
If that’s not a must you could make your reservation from the 13th. Upon arrival in the hotel you could check if they have a room available. Maybe you get if for free as a goodwill gesture, maybe they will charge you for it. If they haven’t got anything available you will still be able to leave your luggage until your room becomes available.
Regarding check-out it’s more or less the same. You can leave your room before check-out time and leave your luggage at the reception. If you want to use any of the facilities of your room after 10:30 you can request a late check-out. Sometimes this is offered for free, sometimes for an additional charge. 2AM is however really late, think of late-checkout as noon, 2PM maybe 4PM at most. So if you really want to leave from your room to the airport your only option will be to include this day in your reservation.