I was wondering about Luke's green lightsaber too. I think it would've made more sense for Luke to give that to Rey rather than Leia's heretofore unknown lightsaber. I suppose he probably chucked it into the ocean shortly after arriving on Ahch-to, though.
As for fixing Anakin's, clearing some time has meant to have passed between TLJ and TROS (given that Finn has been able to grow his hair out), so I would assume that Leia must have helped Rey repair it, yes. A nice nod to it having been broken and repaired would've been to include a welding scar instead of making it look like it never actually broke.