Did the NYC own or control the Rutland?
Yes.
I don't have the exact dates available right now, but the NYC gained control of the Rutland through partial stock ownership early in the 20th century. That continued until the Rutland went bankrupt during or just after the depression, and even then there was NYC influence. The Chief Engineer of the Rutland during NYC control was Luis G. Morphy, who became the trustee in bankruptcy. After the Rutland was reorganized, Morphy became Division Engineer on the Boston and Albany, which was definitely controlled by the NYC.
Some of the Rutland cars in various historical collections have NYCRR brands on the truck frames, and the Rutland motive power practices followed NYC very closely. The F2 class 4-6-0's on both railroads were practically the same design. The NYC's became F12 when they were superheated, but I think the Rutland kept at least some of them as saturated F2's. The details are in Jim Shaugnessy's book.