On the alarm, the original one from the factory does not make any chirping sound when locking or unlocking the car. The only indication you get, is hearing the air pump for the locks like you describe. Also the front and back lights will flash to indicate the car is locked or unlocked.
If you replace the alarm with one of the newer replacements they make, it will chirp when the car is locked and unlocked (from what I read on other threads here).
My alarm also does not work at all. It is completely pulled out of the car right now, and that does not affect anything having it completely pulled (other than the obviously is no alarm). Lock and unlock still works fine from the remote, and car starts normally.
So, other than hearing the alarm when you press the PANIC button on the remote, the alarm is not going to interact with the remote unless your car has a newer alarm module that chirps.