Right!

I searched for the yellow/blue wire underneath the passenger seat. There are 2 of them actually of the same colour, while the smaller plug actually seems to return to the driver's side and not go along the passenger side.

As they were 2 equal wires I decided it would not hurt not to cut them but try to unplug both plugs from the alarm first. I did so - both plugs out. Did a test - opened window and locked the car with a key and then opened the door from inside - the alarm did not go off. Happy happy joy joy. The alarm seems to be disabled. The car starts OK and drives OK, I don't think there are any other (important) devices affected with the disconnected alarm.

I connected the 12 sec time relay (powered off the new alarm box) to the window closing wire and it works! When I remotely lock the doors all windows and roof do close. When I unlock the factory alarm does not go off. This seems to have solved the problem.

Thanks for all your help!

P.S. W you are right - a procedure must be remembered. When locking the doors and unlocking them immediately within the 12 second interval one must wait for at least 12 seconds because when locking again after that, the windows go down instead of up.