Mount Vernon is a city in Westchester County. For specific information about the area itself, you should post in the Westchester forum.
When you commute to Northern CT every weekend, would you need a car? It would be much easier to keep a car in the suburbs than in Manhattan.
As for commuting to your job on the UES from Mt. Vernon, you would take Metro North (east of Hudson line), which is only ~30 minute ride (
www.mta.info/mnr) into Grand Central Station or even better into Harlem 125th Street which is only ~20 minutes.
Mt. Sinai is between 98th and 101st Street on the East Side, so you would take the downtown 6 train from 125th Street (or the uptown 6 from Grand Central (42nd) if you chose to take Metro North down there).
Where are you moving from?