It states in the Shulchan Aruch, chapter 18, that a Jew must not ask a non-Jew to do work for him on Shabbas. However, a Jew is allowed to hire a non-Jew to do work or business for him as long as he does not specify that it be done on Shabbas and does not benefit from it being done then. How then can an Orthodox Shul or Orthodox household employ a Shabbas goy? It seems to me to be forbidden. Could you explain with citations in English?

We may have a non-Jew do work like cleaning dishes, sweeping the floor, set the table, etc. - work that we may do on Shabbat ourselves. We also may have the non-Jew do work for us in general (outside our home or property), if he is not necessarily obligated to do it for us on Shabbat - (source: "Code of Jewish Law" 244:1; MB there - [has been translated to English by Feldheim Publishing]).

We are not allowed to instruct a non-Jew to do work specifically for us on Shabbat, unless it is a rabbinical prohibition, and for the sake of a Mitzva - (source: ibid 307:4-5; MB there).