You'll get the red book when you register at gov facility/ KKI/ hospital. That is your clinic card and used only for pregnancy period
After birth your baby will have an entirely new book on their own

gov hospital will not turn you away incase of emergency. But it will take a longer processing time. they will provide the pink book as well.

Benefits registering in nearby KKI - they provide vacination foc and other consultative stuffs. also with the pink book you can actually make a schedule booking for delivery facilities especially for FPP section.

depends on what you want, but nothing to loose relatively