Some quick he’oros (just in case any of you forget I’m still here : )
1. The Sefer haChinuch writes that the mitzvah of “lo tisaveh” (coveting that which belongs to another) is a protective safeguard; it ensures that a person will not be tempted to steal his/her neighbor’s property. This is not just an exegesis of ta’amei hamitzvos, but creates a nafka minah l’halacha: since theft is prohibited even for a non-Jew, the mitzvah of lo tisaveh applies to non-Jews as well. R’ Yosef Engel in his sefer Lekach Tov questions whether siyagim, prohibitions which exist to safeguard against more severe infractions, are all Rabbinic in nature, or whether a mitzvah or issur d’oraysa can be a siyag. This is a nice example of a siyag s’oraysa.
2. The Minchas Chinuch writes that the mitzvah of yichud Hashem applies to non-Jews as well. The Rama (O.C. 156) famously writes based on Tosfos in Sanhedrin 63 that there is no issur of shituf for a non-Jew, but achronim take issue with this reading of Tosfos. Be that as it may, with respect to the necessity of belief, the Sefer haChinuch writes that even if someone behaves with midos tovos and is otherwise observant, he/she receives absolutely no credit for his/her good deeds absent the foundation of belief. In other words, othopraxy is meaningless without orthodoxy. A footnote in the new edition of the M.C. cites a similar comment of Ramban in his introduction to Sefer Iyov. What is striking about the Ramban is the example he gives – his illustration of a non-believer is someone who believes in “kadmus ha’olam”, i.e. someone who thinks the world has always existed and was not created by G-d. We could get into a whole Moreh Nevuchim / Ramban debate here, but I don’t have time for it.