What was the Diamond Arms 410 with the rabbit engraved on it used for and what is the value of one with a three digit serial number?

Like any other .410, it would be used for hunting small game, like rabbits or dove, or dispatching small preditors like possums or skunks around the hen house. And you don't have a 3 digit serial number - that will just be a manufacturer's batch number or part number. Value will vary by condition, but if it is a single shot, it is worth $35 to $75, or if it is a double barrel it will bring $100-$250.