Answer: Not really. If you were going to damage anything, it would have been more related to how you started the engine after a long unplanned storage, not the old fuel. The old fuel is more likely to cause problems with things like clogged fuel filters and carburetor passages. I find it amazing that in some cases fuel deteriorates to the point of being problematic in as little as two years, while in other cases I’ve successfully started engines with fuel that was over ten years old with no harmful effects.
No matter, if I were you I would consider myself lucky up until this point and get some fresh fuel running through the engine. Drain as much old fuel out of the tank as possible and refill it with fresh fuel along with an additive designed to clean fuel systems. Drive the car for at least fifty miles to assure that fresh fuel (and additive) has run through the entire fuel system. You will probably find that the idle will have smoothed out, and that the exhaust smells normal again. Finally, change the fuel filters as they are almost certainly clogged, at least partially.