why do the planets stay in orbit around the sun

The Solar System was formed from a rotating cloud of gas and dust which spun around a newly forming star, our Sun, at its center. The planets all formed from this spinning disk-shaped cloud, and continued this rotating course around the Sun after they were formed. The gravity of the Sun keeps the planets in their orbits. They stay in their orbits because there is no other force in the Solar System which can stop them.
Depends on what you mean by need.

For any situation which we can expect to produce in a laboratory, or which we expect we might actually find naturally occurring in the Universe, at least one of general relativity or quantum mechanics is adequate for describing it, since at most one of those two theories is relevant. But neither of those theories prohibits the existence of situations where they're both relevant.

For instance, suppose you have a black hole. The theories we have now are sufficient to predict that they'll produce Hawking radiation and evaporate (this does actually involve some mixing of GR and QM, but in a way that's innocuous: You basically assume that the particles are quantum, but living in a non-quantum spacetime). Just wait, and the mass of the black hole will decrease as a result of this radiation.

It'll take an extremely long time for the sort of black holes we expect exist, so we don't expect to ever actually observe it, but it'll happen. Under the current theories, the smaller a black hole gets the faster it'll evaporate, and it'll produce more energetic and more massive particles in the process. Except the current theories don't actually put any limit on this: Extrapolate them out, and you would eventually get a black hole that's so small that it's emitting particles more massive than itself.

This is crazy, so clearly our current theories are incomplete: The proper description must include some way of making emission of such particles non-crazy, or of modifying the behavior of a hole that small so that it doesn't emit particles larger than itself, or of somehow preventing a black hole from ever reaching a size where it would do that.

But any of these expansions to the theory would require a fundamental mixing of GR and QM: You have to have some sort of quantum behavior of spacetime itself in order to achieve any of these effects. Now, just exactly what form that quantum behavior of spacetime would take, well, we're pretty close to clueless on that. But it's got to be something.