Cedar Point & Kings Island help?

so next week I am going to Kings Island (KI) and Cedar Point (CP) for practically the first time. Never been to KI but have not been inside Cedar Point in 13 years. So I had a few questions. I've always been a Six Flags guy so Cedar Fair is not my game.

Would a souvenir bottle from KI work at CP?

Are their coasters that require you to buy a locker to hold personal belongings? (In other words, am I allowed to bring a bag or souvenir bottle to the coaster platform)

If some rides require a locker rental, is there a pass that I could buy that works for all lockers I would use?

Is the Pro Fast Lane worth it at CP on Thursday (week from tomorrow) with a 30% chance of thunderstorms (as of now)?

Besides that I think I have everything else answered. Thanks for your help! Help from one specific park out of the two is appreciated. Thank you!