the flashing light would proberly be for the engine immobiliser which would come on automaticly. this is controled by a tiny computer chip embeded in the key.
you should not need to worry about the alarm flattening the battery unless you leave the car for several months without starting it.
modern day alarms use very low power.
i dont know what its like in Greece but mostly thieves dont care whether a car is in someones yard or not, they will still try to get in.
best to use the alarm all the time.
i dont know much about the eating habits of pet rabbits i`m afraid , but so long as the car dosn`t look like grass i`d think it should be ok.