OP, when you married your wife, was it in the Church or was it outside the Church?
It would really be a good idea for you and your wife to talk to the pastors of both of your churches to see what the congregations you belong to actually believe. Let us hope, however, that when the paperwork for permission to enter into a mixed marriage was filled out for the Catholic Church that your bride-to-be actually read it.
If you married with the blessing of the Church, the bride-to-be ought to have been made aware at the time that to get permission to marry a non-Catholic Christian her intended spouse (that is: you) would have had to reaffirmed his Catholic faith, stated he intended to continue living that faith in the Catholic Church and promised to do all in his power to share the faith with their children by having them baptized and reared as Catholics.
I recommend it be done using water and the trinitarian formula - (in the name of father, son and holy spirit).
If you can’t reach an agreement, do it yourself as the child’s father and thus spiritual mentor.
No. The Church does not condone baptizing a child against the will or without the knowledge of the child’s mother.
This could be a sticky problem, because some capital-B Born Again Christians believe that the only real baptism is actually an interior process by which a sinner repents and embraces faith in Jesus Christ. The sticky part is that some of them believe this process must precede water baptism, which they would not accept as valid if it were performed on an infant not yet intellectually capable of such an act of faith.
This means that if the child attends the Catholic parish church, he would be told that he is baptized, the start of an initiation process that is completed with First Holy Communion and Confirmation (which he does have to give consent to as an adult) but at the other church he would be told that the Catholic baptism did not count.
OP, your bride might learn about the whole process of First Holy Communion and Confirmation and decide that this is essentially the same thing as her church teaches, since Confirmation does require the initiative of the person being confirmed. That person alone pointedly asks for the gifts of the Holy Spirit; parents may not make this decision on behalf of their children.
This is how it is described in the Catechism:
CCC 1285 Baptism, the Eucharist, and the sacrament of Confirmation together constitute the “sacraments of Christian initiation,” whose unity must be safeguarded. It must be explained to the faithful that the reception of the sacrament of Confirmation is necessary for the completion of baptismal grace. ( Cf. Roman Ritual , Rite of Confirmation ( OC ), Introduction 1.) For “by the sacrament of Confirmation, [the baptized] are more perfectly bound to the Church and are enriched with a special strength of the Holy Spirit. Hence they are, as true witnesses of Christ, more strictly obliged to spread and defend the faith by word and deed.” ( Lumen Gentium 11; Cf. OC , Introduction 2.)