If a planet existed in an orbit where the asteroid belt is, would the orbits of all the planets be stable, and what would the climate be likely to be like ?
There never was a planet there. The influence from Jupiter prevented it. Most of what could have been such a planet's mass was accreted by Jupiter itself or ejected from the orbital plane. Some fractions attached to Jupiter's orbit as trojan asteroids.
But, IF a planet existed there, it would not have affected so much the orbits of the other planets. Firstly because of its own low mass, and secondly because most of the orbital dynamics are caused by Sun. It is true that with more than two bodies there is no exact mathematical solution, but numerical models show that the current Solar System is stable (for the planets) in the long run. There are known perturbations caused by some planets in others (in fact, this is how planet Neptune and dwarf double planet Pluto/Charon were discovered).
Such a planet would have a climate in between that of Mars and Jupiter's moons. Think on very low temperatures. According to Wikipedia: "Martian surface temperatures vary from lows of about −143 °C (at the winter polar caps) to highs of up to 35 °C" and "Ganymede's relatively warm surface temperature of 100 K" (-173C).
Assuming there could be a planet in that orbit, could there be any scenario where this planet is destroyed and the solar system reach the state its in today.
The most probable scenario for the destruction of the hypothetical planet is the onw which actually prevented its formation: orbital instability caused by Jupiter. This can cause the planet to be ejected from its orbit to a very eccectric one, to crash into another planet (most probably Jupiter itself) or to fall to Sun, causing extra perturbations to internal planets. By fine-tuning the parameters you'll end up with a system like ours but mostly without asteroids.
Another scenario, alsocaused by Jupiter, would be that the tidal instability caused the planet to torn apart in pieces. This actually happens when a body goes inside Roche's limit for another body, which for Jupiter is smaller that Io's orbit. Note: Jupiter's influence actually prevented the planet to form, but it is probably not strong enough to tear it apart after it would have formed.
Could the destruction of this planet be used to explain the formation of the moon, how early in the formation of the solar system would this have to happen?
In true science, no. Moon being formed by a big impact on Earth is a real possibility, but the size of the incoming body is supposed to be like that of Mars, much bigger than the aggregated size of all asteroids in the asteroid belt. But supposing that the planet once existed, yes, it could have been ejected from its orbit by Jupiter in the very early Solar System, while it was half its accretion process (and thus left some unaccreted debris which are the current belt) and impact the half-formed Earth. In this modified big impact hypothesis your planet would occupy the place of Theia.