Bypass the immobiliser?

I live in The Australian tropics. The car has been in the garage for 2 years avoiding rain storms et al. Its amazing that whenever I open it up after a quiet period I'll find local fauna ensconced in the boot, e.g. marsupials and snakes!

Now I had to swap in a new battery. The "battery man" suggested running the engine while we swapped out the old and fitted the new. That went well. I've started the car up and driven it a few times since the change. However couple of days later I now have erratic alarm/immobiliser problems.

1. The system seems to be slow flashing the lamp on the dashboard but does not appear to be arming, i.e. I can open the door, slide in and fire the engine up.
2. When the system does arm up I can't turn it off with the unlock button. Instead I hit the lock button, the light flashes quickly then I start the engine up before its completed its cycle and can drive the car.
3. Also hitting the lock button the door locks are not activated, either side.

Any ideas on what is going on? Also what is involved in by-passing the alarm/immob system. I'm looking to purchase the EKA code from car-keys-online but that may be of little use if the door mechanism is not working.

Offhand it sounds like the fob should be resynchronized after reconnecting the battery. The process is different depending on the car. Is yours a Mk1 or Mk2 F , a Mk 1 or a Mk2 TF ?.