Since every human child receives half of his chromosomes from his Father and half from his mother, what does the Church teach on how Jesus got the other half of his chromosomes, since he had no earthly father?

There is no official Church teaching in this regard. Knowledge of DNA etc., is very recent and still deepening. Hence one would not expect a thorough theological treatise on a matter of this sort now. 
 
However, one principle must surely apply, namely the teaching from both Scripture and Tradition that Jesus, as a Divine person, also had a complete, intact human nature and was like us in this regard in all things but sin. Hence, he had the complete and proper number of chromosomes. How exactly God supplied the part usually supplied by a human father is not revealed. 
 
Speculation though, always a human tendency, would remain only speculation. We are dealing with miracle and mystery. But this truth remains clear, Jesus, though one person, is fully divine, and fully human.