What is the genetic basis of blood type (ABO) system?
I can answer only half your question.

It is not necessary that the child has to have the blood group of parents.
The inheritance is like this :
1.There are 3 alleles Ia , Ib and Io.

2.Ia and Ib are codominant i.e. they will both be expressed if present together.

3.Ia and Ib are dominant over Io. So, blood group O can only be expressed if the genotype is IoIo. IaIa or IaIo is blood group A and IbIb or IbIo is blood group B.

5.IaIb is blood group AB

So in the case of both the parents being AB blood group,the child can not have O group.S/he can have any other blood group.

The inheritance of negative and positive blood group follows simple mendelian inheritance.

Please feel free to correct me!