I live in Dutchess, have been for 35 years.
Answer to #1 is Yes you can hunt. I hunt on Depot hill rd. Multi-Use land and there are many other state lands to hunt on.
#2 any private land is fair game to shoot all day long on, with few exceptions.
1] noise ordance laws, every town has different times but generally it is ok from 8AM-10PM M-F 9AM-9PM Sat-Sun
2] Must be at least 500 feet from any dewlling or public highway that you do not own or lease.
3] reasonable back stop. ( no shooting towards a neighbors house even though it is 510 feet away.)
Keep in mind all these rules can be thrown out the window if people complain enough and you will lose, so find a spot and make peace with the neighbors and keep the shooting at reasonable times, like not up to 9:59 pm every friday