From what I've gathered:

There has to be a force acting on the Moon to cause it to dip below the Sun since they normally share the same orbital height.

They don't share the same orbital height - at least not always, and not exactly.

The Moon should look larger when it is closer to us and yet it doesn't.

Yes, it would look larger, but since the Sun and Moon are rather small to begin with, and the variations in altitude are not that big either, that's neglegible.

How is it that the Moon and Sun haven't collided?

Why would they collide?

If the Sun and Moon orbit each other, what would ever cause the Moon to dip below and move across the Sun? That would be like the Moon dipping below the south pole and coming up the other side in the round Earth scenario.

The forces affecting the Sun and the Moon are mostly unknown. There's other theories though, for example the Shadow Object passing between the Sun and the Earth.