If Hashem is opposed to Jewish interfaith marriage, why are there so-called “half-Jewish people?” It doesn’t make sense that God would make it possible for a Jew to be capable of impregnating a non-Jew or vice versa.

Your question could be asked as well, why didn’t God make Jews in a way that they immediately vomit any pig, or why did God make it physically possible for Jews or gentiles to have physical relations with animals, or why doesn’t lightning strike down anyone who transgresses a mitzva?! The answer is that life must be full of do-able (!) challenges in order for us to have free-will, and receive the pleasure of succeeding at those challenges, as well as the Godly feeling of independence. In addition, there is no such thing as "half- Jewish". If one's mother is Jewish he is 100% Jewish, and if not, he is 100% gentile.