Within the techum (2,000 amos from the city limits) it is forbidden no matter what whether on Shabbos, Yomtov, or Chol HaMoed
Outside of the techum it is permissible on Yomtov even if there is another city nearby because people don't go out much on Yomtov
Outside of the techum on Chol HaMoed it is permissible so long as there is no other city nearby
Outside of the techum on Shabbos it is permissible even if there is another city nearby because people from a different city are not inclined to suspect someone they don't know of violating Shabbos
Non-Jews built a house for Mar Zutra brei d'Rav Nachman outside of the techum on Shabbos (etc.) in a way that was permissible, yet Rav Safra and Rav Huna bar Chinena would not go into that house, and some say that even Mar Zutra himself did not go into that house, according to one opinion because Mar Zutra was an adam chashuv (significant Torah personality)
According to the other opinion, the reason why they would not enter the house is that Mar Zutra assisted the builders by providing them with straw which made it that what happened was actually forbidden
Non-Jewish waiters of the Reish Galusa are allowed to fix broken tables on Chol HaMoed even if they are getting paid money for their work since the tables are used for eating
It is permissible to verbally negotiate a work contract on Chol HaMoed that will commence after Chol HaMoed so long as it does not entail any normal measuring, weighing, or counting
Whatever a Jew is allowed to do on Chol HaMoed he is allowed to tell a non-Jew to do for him on Chol HaMoed
Whatever a Jew is not allowed to do on Chol HaMoed he may not directly tell a non-Jew to do it for him but he can make it obvious what he wants the non-Jew to do without clearly spelling it out, even if it's something like building a house