Does a shofar need to be made by a Jew? Does the animal need to be shechted according to Halacha for a shofar to be useable? Are there any regulations at all?
A shofar does not need to be made lishma and could therefore be produced by a non-Jew as long as it meets all halachic requirements. It does not need to come from an animal schechted according to halacha but needs to come from a kosher animal (SA OC 586:1 and following, see also The laws of shofar by the OK, under Making of a shofar).
R Yirmiyohu Kaganoff wrote a very detailed and readable article on the topic where he concludes the main issue is that the worker might glue a hole (which is not kosher) or at the extreme that the shofar is made of plastic. For this reason, he recommends checking that the manufacturer of the shofar is a halacha-abiding and knowledgeable Jew. There are various hechsherim that assure buyers of this.
HaSofer Moshe Flumenbaum (a well-known reliable owner of a tefilin/shofar store in Jerusalem) writes here from the perspective of a vendor. He absolutely recommends a hechsher to ensure the shofar comes from a kosher animal, has no plugged holes and was not treated with any glue, lacquer or other coating. He writes that most of the shofars produced in Israel are actually made by non-Jews who might have an incentive to hide defects. Since the largest market for shofar today is actually the non-Jewish consumer, a perfectly looking and sounding shofar might actually not be kosher.