Tom
Here is my try at your questions.
Of the Tillie course that have not been in the national spotlight, there a number that come to top of my mind in no particular order: Binghamton, Alpine, Bethpage Red, Swope Park Muni, Philadelphia Cricket, Kansas City, Mission Hills, Golden Valley, Oak Hills, Rochester, Brook Hollow, Sunnehanna, Niagara Falls, Scarsdale, Belmont (FKA Hermitage). Of these, I have only had the good fortune to play Alpine, Bethpage Red and Binghamton. The others and many more are on my to play someday list.
As far as his greatest influences, Tillie was was clearly his own genius, and only he could truely answer this question. My speculation would have to consider all of his life's experiences to include:
His many visits to Scotland.
Old Tom Morris and many of the other Scottish/British designers, like Harry Colt.
His play over some of the other early great American courses like NGLA, Garden City and Merion
His hand in Pine Valley.
And his early laboratory at Shawnee-on-the-Delware, as he redesigned several holes there, perhaps some by trial and error.
In addition to the Sahara or cross hazard common to par-5's, I often see the following Tillie features in his courses:
Routings with an outer and inner loop.
Contoured approaches into greens providing for the ground game. But not on every hole.
Master guarding pits on one or both sides of a greensite.
Deception bunkers fronting an approach into a green.
The incorporation of natural features into the strategy of play -- greens on natural hillsites, greens next to water hazards, etc. (Personally, I think Tillie would abhore the common waste of a water hazard that we often see when a green has been constructed with a pond fronting it, but the pond is 25 or more yards short of the green with a bunker or two between the pond and the green -- the pond has been wasted)
Elongated/drawn out green side bunkers that extend into the approach to the green.
Elongated and angled fairway bunkers.
Relatively steep faces on green side and fairway bunkers, similar but not as severe as the common Scottish links bunker. These bunkers were built by hand with shovels -- providing the detail of hand-sculpting, which is generally not seen in the the common American bunkers shaped with even the smallest of bulldozers.
In regards to Tillie and Alistair MacKenzie, based on my current knowledge, I would go so far as to speculate that Tillie was very competitive towards Mac and did not agree with certain aspects of Mac's design style such as Mac's edging of bunkers.