Baptism is a sacrament of the church given to the church. As such it is our responsibility to administer it unto those whom Christ has said to administer it unto. As such, only those who are members of a local body (by profession or birth to a professing family) may receive the sign since it is not administered apart from those who are in Christ but administered in and by those who are in Christ. This much is assumed in the Belgic Confession (BC, Article 33) & Heidelberg Catechism (Q&A 74) when they state that baptism is "by which we are received into the Church of God, and separated from all other people and strange religions, that we may wholly belong to Him whose mark and ensign we bear" and an "ingrafting into the Christian Church".
So no child should be baptized whose parents are not members of the church, nor should children be baptized if their parents are under discipline (see HC Q&A 85).
With respect to the second question, I would lean towards 'yes' if they are baptized in a Christian church but, at least as our (Reformed churches) form of baptism goes, the parents really wouldn't be allowed to have their child baptized because the questions assume faithful membership and godly living in the ones who answer them.
However if the church administers the sacrament unfaithfully the child is still baptized, whether we like it or not. Perhaps the parents and child have been unfaithful as well but that does not undermine the legitimacy of the sacrament itself because it is, in the end, it is what God promised, signed and sealed unto us that makes it valid.
