Where these sorts of essentially non-naturally-occuring materials are concerned, invent probably is a better choice than discover, I think we usually do say 'invent' with stuff like plastics, for instance. You pretty much have to invent the process for making them to begin with.

As for space elevators, no not really, long tethers are all about tensile strength relative to length, now the ability to use such a configuration with something like carbon nanotubes or graphene might come in handy but elevators are all about how well a material does when you hang a big weight suspended from it, at that scale, basically can a yarn-width piece of the material suspend an aircraft carrier from it? And we'd have to see how the material stacks up in terms of tensile strength. I haven't seen a listing for that yet, just how it handles compression which is quite amazing. I will cheerfully stand corrected if the stuff turns out to have a TS of tens or hundreds of GPa of course.



Well, better batteries have all sorts of advantages but not really for space travel, unless they allow significantly better than 100 MJ/kg. Batteries are of interest for moon bases, because of the long dark periods, but generally in space outside of earth orbit the Sun's always shining and a good deal brighter than planetside, where you'd usually dump extra mass into redundancy or extra solar panels. There's certainly no downside to space travel from lighter batteries, as mass is everything, but in terms of space batteries are mostly important for satellites (when the Earth blocks the sun), and that's more about endurance and lifetime though obviously mass is an important factor in initial launch cost, I don't know what the actual battery mass to total mass ratio generally is in a communications or surveillance satellite but my SWAG would be less than 10%.