Yes.
Yes. A person baptized into the Catholic Church, or received into the Church by a profession of faith is always Catholic even if they cease to practice the faith.
Catholicism is not something “done” to anyone. Catholicism is a *relationship *with Christ. That relationship begins with Baptism. If the person is under the age of reason, the parents ask for baptism for the child and promise to raise the child in the faith. If the person is over the age of reason, they ask for baptism themselves.
Yes, baptism should be confer upon infants without delay. It is a very serious obligation of all Catholic parents.
A child coming to the Church over the age of reason must state their desire for baptism.
Perhaps reading about the Sacrament of Baptism would help:
vatican.va/archive/catechism/p2s2c1a1.htm
From the Catechism:
1250 Born with a fallen human nature and tainted by original sin, children also have need of the new birth in Baptism to be freed from the power of darkness and brought into the realm of the freedom of the children of God, to which all men are called. The sheer gratuitousness of the grace of salvation is particularly manifest in infant Baptism. The Church and the parents would deny a child the priceless grace of becoming a child of God were they not to confer Baptism shortly after birth.
1257 The Lord himself affirms that Baptism is necessary for salvation.