Hello let me help you with any of your Honda problems.
It sounds like you are talking about the security system that is giving you the problem......the immobilizer is the system that will allow the car to crank but not start if a registered key is not used.
It is very possible that the keyless remote is causing the problem as those remotes were known to have bad keyless modules that would cause them to function intermittently. There are two ways you can approach this......one take the battery out of the remote and see if the problem stops.
The second thing that can be done is have the dealer take a look at the breach history using the Honda scan tool. This history will tell them what component caused the alarm to trip and that will lead them in the direction of what needs to be checked.
Now you also state your battery is dead and could they be related......yes a weak battery can cause all kinds of electrical bugs including the alarm sounding by itself. I would start the whole process by having the battery tested and then replaced if it will not hold a proper charge.