the key is going to be a bit pricey.. 180-200 range from dealership only. You'll need proof of ID to car. 10 days usually. Not sure if you can replace battery-that model may allow.
Dealership reprograms the key to the car from my understanding. unless there is another means?
As for your door lock -never force it!!
the cylinder sounds like it's spinning in the lock which means something broke free. if you haven't used the lock for a time and it's very cold - heat the key up with a lighter and then insert into the lock. that usually melts the ice inside and problem fixed.