No. Even at the peak of the cold war when both the US & USSR were at their largest stockpiles there was not enough.
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No, there are not enough nuclear weapons in existence to knock Earth off of its orbit. The force required to move a massive object like Earth would be astronomical and far beyond the capabilities of any known weapons. Earth's orbit is stable and not easily disrupted by human actions.
No. Solar flares are a common occurrence. Earth's magnetic field protects us from the worst of it. The greatest danger is that an especially powerful solar flare could knock out satellites and damage electrical systems.
Earth orbits the sun because of the balance between the gravitational pull of the sun and Earth's velocity. Earth's motion prevents it from falling directly into the sun, keeping it in a stable orbit.
The presence of space junk affects the operation of TV satellites, weather satellites, the International Space Station, and any spacecraft that's launched from Earth, because a nut or a bolt hitting one of these at a zillion miles per hour can knock them totally out of commission.
Yes, the Aurora borealis, also known as the Northern Lights, emits light when charged particles from the sun collide with gases in the Earth's atmosphere. These collisions produce colors such as green, red, purple, and blue that are visible in the night sky near the polar regions.
It's really impossible to say without knowing how the Moon was destroyed. Assuming it was done in such a way that none of the pieces hit Earth, there wouldn't be much of an effect on Earth at all. There wouldn't be any more eclipses, tides would be smaller (and occur at about the same time every day), and nights on average would be slightly darker. If some of the pieces did hit Earth, it would be the mother of all meteor showers.