I've returned home after a trip to my sister's and I read some confusion a fan has. So if anyone else is confused let me clarify.
SPOILERS AHEAD for Hal Jordan and the Green Lantern Corps
Yes, Booster kind of helped for the Sinestro Corps.
No, Rip Hunter isn't a Green Lantern. Future John Stewart just gave him a ring that can only access information and Rip stated he wasn't chosen. Since he's unknown to most a ring would be a pain in the rear anyway.
By "daughter" I assume they means Rani although she was more of Michelle's adopted kid than his (Booster was always away) although she seems to have disappeared. It was also hinted that her family was actually looking for her anyway so Rani probably wouldn't have stayed. Anyway Rani wasn't a Daxamite. Her foster family was visiting the planet.