Transit at Frankfurt

Hi all. First time taking a transit flight booked through SIA. Flying from SG - Frankfurt, then onward to Dublin via Aer Lingus. Layover time about 2 hours.

Very noob question as I’ve always flown direct and first time transiting - are the check in luggages tagged for onward loading to the Aer Lingus flight? And same for my return trip? Do I have to leave the terminal and re-check in for the onward flight?

Also just to sidetrack, is hairspray below 100ml allowed in carry on bags? Thank you 🙏🏻