I have a couple of questions:
1. Spearmint plant in 3″ pot getting lots of black spots and dying…what is wrong?
2. What is ideal watering in s window in NYC?
3. Using pebbles with water that you suggested, what is the water level in the tray relative to the pebbles; does the pot get any direct water on the bottom?
A 3" pot is quite small and is likely a starter pot.
You may need to repot the Mint, since it does grow quite quickly.
Here is a link with care and growing requirements.
The spots could be do to overwatering. Also if you have treated water it can cause the tips of leaves to darken. Try leaving water to sit 24 hours before watering.
If the spots seem to be spreading, it could be a fungus. Treating with Neem Oil will help. Neem Oil is safe for people and pets and is safe to use on herbs.
Here is a link with more information.