Ever since I bought my 98 ex coupe my cruise control hasn't worked. The button lights up when you push it in but it does not hold speed or do anything when you press the buttons on the steering wheel.

The second problems is when my lights are on and I open the door there is no alarm to shut them off. Is it the little black switch on the door that is broken?

 

Check whether the rubber stopper that presses on the cc switches at the top of the clutch and brake pedals (stopper for switch B) has fallen out or broken.

The lights-on reminder is part of the integrated control unit plugged into the rear side of the dash fuse box.

Ignore red arrow in this case. It's the stopper for switch B that you are interested in: