Q. If a Goy builds my suka and also puts the schach as instructed by me. Do I have to lift the schach a tefach or 3 to make it kosher lechatchila?

A. Shulchan Aruch (O.H. 635: 1) permits a Suka that was made by a Gentile as long as the intention of placing the covering schach was for providing protective shade. However, Poskim write that on the onset a Gentile should not build the Suka, if it can be avoided. (Piskey Teshuvos ibid. quoting Hochmas Shlomo, Mishna Berura, Kaf Hachaim and others who maintain that we should lechatchila be concerned with Rabenu Tam’s opinion on Gittin 45b.) Sdei Chemed (Ma’areches Suka 2: 3)writes that one should be more stringent on the schach than on the walls.See also Nitey Gavriel (Sukos 25 n. 9) who quotes a number of Poskim that a brocho should not be recited on such a Suka, if the schach was not raised.
Horav Shlomo Miller’s Shlit”a opinion is that you don’t have to lift the schach once it has been placed on the Suka by a Gentile following your instructions.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a.