I don't know the exact details but my brother (British Citizen) and his China Chinese wife were able to obtain British Citizenship (passports) for their kids. I think they had to send the original copy of marriage certificate.
3...If your child doesn't have a British passport than he doesn't have 英國國藉.
In the UK, you will find you get asked to submit your child's birth certificate when you register with a GP, apply for school, and child's benefit. If they see your child is not born in the UK, they would want to see that he has a British Passport.