1) They meant there's no way to get negative energy particles which could be the source of antigravity. That would destabilize the vacuum because it would allow you to keep lowering the energy without bound.
2) Intuitively if antimatter is regular matter travelling backwards in time, then at the start of time, wouldn't it all go the other way? Kind of like one huge quantum fluctuation.
> if antimatter is regular matter travelling backwards in time
It isn't. More precisely, while the mathematical trick that this phrase refers to (which also, btw, requires reversing all charges and flipping parity) works as far as making calculations simpler while still getting the right answer, it doesn't correspond to anything physical.
2) Intuitively if antimatter is regular matter travelling backwards in time, then at the start of time, wouldn't it all go the other way? Kind of like one huge quantum fluctuation.