Any of the NYC area Airports. Newark Liberty is however the closest.
Yes and No. It is within walking distance. However I recommend taking the train to Convent Station in Morristown and taking an Uber. The train to Convent is more reliable and faster from NYC. If you do prefer to take the train to Bernardsville I recommend you attending the Reception only. The walk to the restaurant is well paved and only half a mile.
Don’t stay at the Old Mill, it’s close to our house but over priced and the rooms are very dated. There are several hotels in the area. But if you plan on hitting NYC as a major point of your visit definitely stay in Jersey City or NYC. You’ll have better access (Obviously you’ll have better access to NYC in NYC. Jersey City and Hoboken by the path would be more roomy and affordable to NYC)