Am I responsible for someone falling on sidewalk adjacent to my residence?

The answer to this question is going to depend on the location of the sidewalk and the ownership of the property on which the sidewalk is located.  In many areas, the sidewalk is actually located on your property, and the town in which you live has an easement over that part of your property.  This makes the sidewalk your responsibility to maintain and can make it your responsibility if someone falls on the sidewalk.  You might start with a survey of your property to see the location of the sidewalk, and also talk to the zoning department of the town in which you live.