As said in the earlier reply, you can use ME but you will have to collect your own luggage and bring it on the motorcoach. Per Disney's site: For airport arrivals after 10:00 p.m., Guests must collect their luggage at Baggage Claim and transport it with them on the motorcoach.
No, you will not be able to check into your room for the night. It may technically be the next check in "calendar" day, but the hotel guests that checked in before midnight (or checked in earlier in the week for their stay) are allowed to remain in their room until check out time (11 AM). So if your assigned room is still occupied with guests, you can check in at the front desk of the hotel, but you'll have to wait until Disney calls or texts that your room is ready. If it wasn't occupied you might get lucky, but the odds are probably against it.
The safest thing to do is book the extra night at your resort and call Disney to link the two reservation numbers together in the hopes that you won't have to move rooms after your first night. Getting a hotel at the airport works too because if you are relying on ME for transportation you have very limited options: extra night at Disney hotel, airport hotel, or hotel close to airport with shuttle and take ME next morning.
Have a great trip and good luck!