By the power of truth, I, while living, have conquered the universe.
The Crystal wasn't confirmed to be Forerunner, I'd speculate it would be a Precursor artefact because it just seems to be too advanced for it to be something the Forerunners made.
But I might argue against that by looking at Cryptum where Faber creates several crystals by chiipping them away from the Core Crystal. So whilst it's possible the Forerunners manufactured it, it's not proven that they created the crystals.
The Keyship should have had an intact slipspace drive, the ship itself wasn't damaged once during the battle against the Sangheili and since the formation of the Covenant and High Charity it hasn't moved. In Contact Harvest, Mendicant Bias attempted to pilot the Dreadnaught and go to Earth to take humanity to the Ark, how could he do that without a slipspace drive?
I see what you're saying and it makes perfect sense, but I also see a fair bit of evidence to the contrary.