This is not a crazy question. I have thought about this multiple times as well. Here is the problem: We have absolutely no idea what the answer is, and we have no possible way to answer the question, so this one will just have to be one of those unanswered questions!!! What we can say for sure is that Jesus did have 23 pairs of chromosomes. It is reasonable speculation that 23 of them came from Mary. Jesus surely was of the same race as his mother, otherwise that would have been noted by his contemporaries, and it was not. But this is nothing more than reasonable speculation. One thing we know for sure is that Jesus had a Y chromosome, and he did not get it from his mom. You ask whether Jesus had the Holy Spirit’s DNA. The answer is no. The Holy Spirit does not have DNA. The exact nature of the Holy Spirit is an interesting question, but we can be sure that he does not have DNA.
John the Baptist told his hearers that God could create sons of Abraham from rocks. Jesus created fish with the proper DNA that people ate. Therefore, we can be sure that God could do whatever he liked in creating the human being Jesus of Nazareth. Whether all of the chromosomes were created from scratch, or he used 23 from Mary, we will have to be content with not knowing the answer.