There seems to be no exact method to force the cat to climb down the wire to the ground floor. Some players simply block the cat NEAR the shaft and wait for it to jump on the wire.
If you click on the cat while there is milk downstairs, it will start an automatic chase routine, after which the cat goes down the wire.
You have to bring the milk to the cat. Go to the farmhouse left of Main Street, enter through the chimney, and get the cat's bowl from the downstairs floor. Take it to the diner and fill it with milk.
The cat's bowl is downstairs in the deserted farmhouse. If it isn't on the first floor, it is already in your item bag.
The cat's bowl is in the deserted farmhouse. Take it to the diner and have the waitress fill it with milk. Bring it back to where you found it, and chase the cat from the tub to get it downstairs.
Go to the farmhouse left of Main Street, go in through the chimney and get the cat's bowl on the downstairs floor.
There is no attic on 24 Carrot Island. You can climb the electric cord to the right to reach the second floor bathroom in the farmhouse. Turning on either knob in the shower will chase out the cat hiding there. Once you have a bowl of milk for the cat downstairs, chase her by clicking on her, and she will go down to drink the milk. (see related question)
"Charlie" is the owner of the hardware store "Charlie's Carrot Surplus". It is never stated if this is the woman there with the lost cat, or if she only works there.
The cat (Whiskers) belongs to the woman at Charlie's Carrot Surplus (hardware store).
Whiskers, the lost cat, is gray.
The waitress at the Carrot Diner will fill the bowl for you.
Take it back to its owner at Charlie's Carrot Surplus.
On 24 Carrot Island you don't get a wrench. But you do get a pryer thingy. You get that by returning the hardware lady's cat to her.
The cat belongs to the woman at Charlie's Carrot Surplus, and is hiding at the farmhouse. You will receive a reward for the cat's return. (see related question)