Mind you, Eleven, I'm no physicist. But I'll give it a shot.
An object travelling at 100,000mph would pass by very quickly so it's effect would be minimal. If it passed halfway between the Earth and moon, it would be 125,000 miles from Earth. The Earth is only 8,000 miles in diameter, at that speed it would travel that far in less than 5 minutes. Possibly the effects of it's gravity could affect the tides and cause some atmospheric disruption. But since it would pass so quickly, not much of either.
If the moon blew up and shattered into 4 or 5 pieces... well that's a tough one. It depends on how it blew up. If the explosion was very powerful, those pieces would follow new trajectories. The chunks that were blown in the direction of the Moon's motion would either leave earth's orbit or find new equilibrium in a higher orbit. Those chunks blown in the opposite direction to the Moon's motion would either fall to Earth or find equilibrium in a lower orbit. The chunks blown directly out from the Earth or directly toward the Earth would end up in highly eccentric orbits, some of them would undoubtedly hit the Earth. Right away or even after many months or even years. That goes for huge chunks as well as tiny pieces.
If the explosion was not very energetic, the Moon might crack apart , but then eventually the pieces would drift back together under their own gravity.
This is my best guess. I think there are other members here that have better understanding of these things and so could give you better answers. Good luck with your story.