Have take the car to a Honda dealer to check on a service bulletin regarding door lock actuators. If they haven't been replaced via this service bulletin, the dealer will be able to.

 

Yes. Replace the door lock actuator on the driver door. If the other door actuators are functioning this will solve the problem. Now, you can lock all the doors upon exiting the vehicle if you pull and hold the exterior driver door handle out with the door open, push down on the lock on the door, not the power lock button, and it will lock the driver door; and if the other actuators are working, this process will also lock the other doors with power locks. With the faulty driver door actuator, the only way to unlock the door if the fob doesn't work, is with the key.
Source: happened on my '03 and I replaced the driver actuator and the fob worked again.

 

The car driver's door won't lock manually (pushing the button) if the car door's open, only closed.

this part of your post is normal. if you want to lock your car door, while the door is open, you will need to pull the outside handle at the same time. I’m certain that this must have prevented absent minded people from accidentally locking their keys inside the car while they are outside of the car.

other than that, yeah, things are pointing at your door actuator as the issue. Make a note, as Japan and UK made 2003 CRVs have different actuators. Having a VIN number when ordering the part would be beneficial.

having changed both a Japan and a UK actuator, the Japan actuators are more of a pain to do. in other words, if you have a UK - you can probably change the actuator itself inside the door. For a Japan made car, I found it easier just to remove the whole latch, and deal with it outside the car.

 

Alright ! Thanks for the advices everyone, I'll order the part and do it myself.