White PVA wodworking adhesive is probably as good a glue as any other.
No you will not be able to buy hardwood in the sizes you require - someone might make it for you from larger stock by cutting and planing but it will be very expensive.
Fitting new slats into existing mortice slots will be difficult-to-impossible without dismantling parts of the existing seat.
Given its age you may consider just buying a new seat - to repair it will be a labour of love - lots of effort.
Good luck...
Peccavi
May 2010