No, God doesn’t have gender.
Gender is a complex mix of biology and culture. It tends to refer either to something that is primarily biology or to a set of characteristics that are thought of as masculine or feminine because they tend on average to be more common for those genders.
I would argue that God has no biology, and that there are Biblical reasons to see in him the best characteristics typically associated with both genders. Both men and women are made in God’s image. In Christ there is no male or female.
Why is God portrayed as male? Of course he isn’t always. Wisdom (which is the basis for the Logos) is portrayed as a woman in Proverbs. But normally he is. I suspect that has to do with the culture out of which Israel arose. God and goddesses tended to have different characteristics, and the Lord was closer to the first.
But I don’t think we ought to push that tradition very far.
I think there needs to a limit to how we use the communication of attributes. How tall is the Father? What color is his hair? Surely Jesus had these attributes.
Firstly, communicatio idiomatum, where limits have been applied to it, and these limits are not universally recognized, is generally applied only in Christological contexts, so we can’t use it to discuss Patrology or Pneumatology except insofar as they are consubstantial with Christ, and also with us, via the hypostatic union, which deifies us to the extent of theosis (participation in the divine energies) but not apotheosis (participation in the divine essence by being made into God the Trinity. The essence/energies distinction is related to the next commonly held limitation on communicatio idiomatum, which is that it does not apply to the “nulas,” that is to say, logical problems arise if we use it to apply Divine immutability and invincibility to the human nature of Christ, or to apply mutability to the Divine essence, but these problems are not insurmountable.
Others go further and draw the line at the “omnis”, omnipresence, omnipotence, omniscience; omnipresence is again logically challenging, and this challenge is confirmed by a seeming indication of locality in the hymnography of the Byzantine, Coptic, Syriac and Armenian rites, which of that translated seems to go into the most detail concerning the nature of the incarnation (whereas the Western Rite hymnographic tradition seems to reflect more on what this means for us, and the emotional response, as is demonstrated by these paraphrased Eastern Orthodox and Protestant lyrics from the evening service on Good Friday: “...Today He who is unbound and the creator of all is sealed in a tomb...” and “Were you there when they put him in the grace? Sometimes it makes me tremble, tremble, tremble...”
However including omnipotence and omniscience makes no sense from a Chalcedonian or Miaphysite standpoint and I am not convinced of a lack of divine omnipresence in the person of Christ.
Earlier in the conversation I reviewed and debunked the idea of the Patriarchal culture of Israel having an impact, and I lament to see this point raised.
I also, given the unknowability of the divine essence, would note that your statement “God has no gender” cannot be taken in an absolute sense, since you have no way of knowing; your specific example of Proverbs as an instance of the energies of God being genderless presupposes we accept your interpretation of it, which I feel is intriguing but inaccurate, an attempt at Alexandrian metaphorical exegesis but one that ignores typographical and metaphorical aspects, or else, you applied Antiochene literal-historical context to one of the two books of Holy Scripture where that approach is least edifying (the other being the Song of Solomon).
I will instead propose that a more Alexandrian interpretation is that the esoteric meaning of the Song of Solomon is what some Greek Fathers and saints actually called an erotic love (devoid however of lust or concupiscience) between the Church and its Savior, and in Proverbs, the feminity of Wisdom is a typological prophecy that Wisdom would be carried by a woman, literally, the Blessed Virgin Mary, who bears Wisdom and is the means of the incarnation of Wisdom and who in my belief and those of many Christians mystically leads the choir of saints as she intercedes on our behalf to our divine Intercessor.
But thank you for as ever an interesting and challenging reply, and I really would like to have a deep dive with you along the themes of the undiscovered Calvin, or Calvin vs. Calvinism, if that interests you, and I sincerely hope it does.
God bless you!