suppose the moon were held in its orbit not by gravitational interaction with the Earth but by a long, mass-less cable attached to the center of the earth. What would be the tension in the cable?

Answer 1

What is centripetal force? massmoon*v^2/r

But I wonder what would hold Earth?

In fact the moon is not orbiting around Earth,but around a common center of gravity. That probably is too compliated for this thinking, after all, a massless cable almost 200,000 miles is a stretch. Pun intended.