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My las tiles are all in a coordinate system that uses US feet, so if the res parameter for grid_terrain() is set to 1, is the output 1-ft resolution? I figured this is the case, but my professor wondered if the function only uses meter units, in which case the output is 1-m resolution.

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Yes. In lidR, inputs are unitless. The software only makes stupid comparisons between the coordinates of the points and the input values. So if you ask for grouping by 1 unit it is 1 unit of the point-cloud. It works the same if you want to query point above x or anything else. The CRS of the input point-cloud is always propagated and preserved in the outputs.

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