An illegitimate child can be baptized.
yes the child is only once baptized.
No, the child needs to be baptized. Catholicism is not passed by generation, rather by the Sacrament of Baptism.
Yes, a Catholic child should be baptized in a Catholic church.
Any child may be baptized if there is a reasonable hope that they will be brought up Catholic or in danger of death.
Yes
yes if the child accepts Jesus as the savior of his/her life.
Yes, a child can be baptized if his parents are separated, even if they are divorced. Children are not penalized for the mistakes of their parents.
Yes.
Yes people do give gifts to a newly baptized child. Even though he/she is unaware of the situations around, but it serves as a memory when they grow up.
Yes, anyone, even an unbaptized person can baptize in emergencies. But if it is not an emergency then the child should be baptized in the Catholic Church. If the Church denies baptism, then it should not be done.
No, a child does not have to be baptized on Easter unless they would like to be.