I think we don't have to rely on the battery of the watch, because if you had noticed the watch was deadnot working, when a grown up Murph took it up. It was not updating time. It was Cooper who was making the second needle move. On how the watch generated data, I think Cooper was using the second needle to transmit data in Morse code.
Morse code is a method of transmitting text information as a series of on-off tones, lights, or clicks that can be directly understood by a skilled listener or observer without special equipment.
On how he was able to use gravity to move the needle, I think although gravity is indeed generated from the center of the earth, still there some value of gravity along all the three axes. I am not an expert on physics, but I think since Cooper could use gravity, it would not have been impossible for him to apply some force along the x-axis. Apart from that, we also notice that gravity is used to disperse sand on the floor and make patterns out of it. So pretty much the same way with clock's needle.
However, this point But how exactly he could use gravity for making movements along the x-axis is beyond me. May be better explained by someone who has goodwith a deep knowledge of physics could explain whether this is possible. Anyway, we might ask how did he get hold of gravity all? But since according to the story, he could, it doesn't seem so hard to assume that indeed he could make a movement along the x-axis, using gravity.