1. If you have any space industry, the space elevator makes it way easier to do things. For example, it's about 15 km/s of delta-v to get to the Moon from Earth's surface, of which ~9km/s of that is just to get into LEO. The space elevator slashes the required amount of delta-v to go anywhere since, as the saying goes, once you're in orbit you're halfway to your destination. If you have no space industry to speak of, you don't know how to make rockets and nothing changes. Processing asteroids in space is doable but difficult since you also have to address life support if using humans for some reason, but otherwise robots can do it fine if your nation knows the laws of orbital dynamics well enough to plot intercepts and captures and such.
2. No negative energy, FTL breaks the future-modern setting just as surely as aliens would. And it isn't one of the only known ways, it's a loophole in the math describing relativity and isn't actually possible, since the Standard Theory doesn't allow negative energy. Antimatter can be created in particle colliders in ultra-small quantities at, I'd say, 175-200 science (particle colliders are huge and serve 0 purpose besides theoretical science, no nation without a significant science investment would make one), but keep in mind antimatter is currently $62.5 trillion per gram and we don't know how to stabilize it so it doesn't react with matter immediately and vanish.