What is the Halacha and the Minhag regarding Non-Jews in Shul choirs at religious services led by a Cantor?
Choirs are rarely found in Orthodox synagogues nowadays (for reasons beyond the scope of your question), as such there is little literature addressing the topic from an Orthodox perspective. However, choirs seem more common in Reform and Conservative congregations and their rabbinical bodies have documented their views on your question. Here are some representative quotes (extracted from R Baruch Frydman-Kohl's Teshuvah: Non-Jews in Synagogue Choirs)
From the CCAR Responsa Committee, the Reform's movement central rabbinical body
The phenomenon of non-Jewish choristers is on its way out. It represents a phase of Reform history which no longer can serve as precedent for our teshuvah. The shelichei tsibbur must be members of the covenant community and they cannot yield this responsibility to outsiders
From the Committee on Jewish Law and Standards (Conservative)
Unanimous opinion of the Committee that the practice of having non-Jews in synagogue choirs is not in keeping with Jewish standards (1954). Gentile choir leaders at services are deemed contrary to the spirit of Jewish tradition and practice, although there would be no objection to using them in training a choir (1958)... Remarkably, for a Committee that prizes diversity and often exhibits disagreement, this decision has been reviewed four times with the same conclusion: choristers must be Jews.
See the original for more history, details and the author's personal conclusions.