Hello, Welcome to Just Answer.
If your corporation was moved to nassau county and maintians a office in Nassau county then you will not be required to file a NYC 3L corp tax return unless the corporation owns real propert, has a location in NYC, or has an investment or subsidary in NYC.
There is no different tax return for operating in Nassau county, you will essentially save taxes
from paying NYC taxes
yes, any presence in NYC will forfeit you the tax brreak
No you will not be required to file a NYC 3L if your just conducting work in NYC.