Mark,
You asked, "Is the property flat? How did Tillinghast address this design problem? Are antecedents and inspirations for Winged Foot to be found there, either today or originally? As his first course, did Tilly establish any design elements or themes?"
The course had several holes on the mainland side by the Buckwood Inn with the bulk of it rambling across the Binniekill on the island. Though the land seems flat in a number of photos from the day, it really did contain many gentle rolls and slight elevation changes, thus allowing for a number of the greens to pitch at angles.
No, I wouldn't say that it served as an inspiration for Winged Foot or other of his great works because, as a designer Tilly didn't think that way and even from this first course he simply designed and routed based upon what the land gave him. It is in that sense that it serves as inspiration for his later works. The par-threes were quite distinctive, especially the famous "Binniekill hole" with the green on the other side of the water. It was most dramatic for its day.
Part of the unique aspects in this designthat set it apart from other designs and made his work immediately noteworthy was how he used features from the UK designs he enjoyed years before. Yet these weren't true "template" type holes, something that Tilly took great pains to avoid throughout his career. For example, there was a wonderful punchbowl and Shawnee was the first course in America to use "mis-Surrey" style moundings on a number of holes.
Unfortunately, when it was expanded to 27 holes, a good portion of Tilly's work was destroyed. We are hoping that in the latest round of work that is being contemplated that the ownership will give strong consideration to restoring as much of Tilly's work as can happen...