I think something is wrong with your shift lock.
The way its supposed to work is when you have the car in any other gear but park, the key won't come out, and if you don't have your foot on the brake, the car won't shift out of park.
It sounds like this mechanism is messed up.
The deal with the car alarm is probably that the thing thinks its armed and you can't disarm it. Take the other key, go outside the car, and lock it with the key, and then unlock it. This should disarm the car if the remote isn't working.
When you start the car and press the brake pedal, do you hear any kind of "click" sound near the shifter? This is the shift lock, unlocking the shifter so you can move it. If you don't hear anything, this may be the problem.