Looks like you have an Opuntia, they drop pads for a pastime. You could save time by taking it off, let it rest for a week or so to form a callus and then pot it up in a mix of compost and coarse grit.
Be careful not to touch it with bare hands, Opuntias usually have "glochids" tiny spines that detach and become embedded in your skin. Use a cloth and dispose of it afterwards.
Once a month is fine for watering during the summer, stop watering till spring now. If you can move it somewhere cold for winter that would be good, but if it has to stay in a heated house then occasionally check that the roots don't dry out completely.
Start watering properly again in the spring and don't worry if it shrivels up over winter