Upon further examination (following clues provided by y'all) I realized that what I had thought was the bracket eye for an arm mounting bolt on each door was, in fact, the hinge check arm itself, pushed back into the door! Pulled 'em out, cleaned 'em up (well, I will clean them, but I was standing in the parking lot at work...), and snapped them in place.
Jack, the spring catch on each door jamb bracket is in good shape, so I won't have to replace those.
Whew! It turns out that the hinge check arms are spendy (according to a Ford parts online website). And I still don't really know how to remove door (or any interior) trim, so I've been saved that learning adventure.
Thanks to Jack, Stump, Carringb, WVvan and others.