Gravitational waves move at the speed of light, so it's hard to make a well out of something that moves. Although it's not impossible given the right type of boundary conditions, but none of those would involve a single object moving about a point. It would be a continuum of matter arranged in such a way that you might be able to produce a standing wave.
No. Gravitational waves travel at the speed of light, and that's the maximum speed your object can go as well. Pi is 3.1415926......, so by the time your object has gone around the circle once, the gravitational waves have already moved more than three times the diameter of the circle.
Now, admittedly, in a closed space the ratio of the circumference of a circle to its diameter is less than pi (just think of the equator of the Earth versus the distance along the prime meridian from the equator to the pole). However, if you're in a closed space like that, it means that you are already in a very deep gravitational well, and if space is really closed, then you are inside of the event horizon of a black hole, in which case you won't keep the moving object out of the very center of the well.