The three body problem you describe has to do with Gravitational attraction and the exact solution to it's equations of motion. There is no exact mathematical solution to this problem, though it can be numerically integrated with not problem.
Three balls colliding off each other has no math difficulties and from my experience with 3DGS PE, it's not a problem within the game.
All the other issues you mention are classic issues of MMOG design, not just focused around the physics engine. Do some research on MMOGs and Multiplayer Optimization and it might answer alot of these questions.