Yes, as long as your hotel isn't in a pedestrian area. In Positano and on the Amalfi Coast, there are many hotels that can only be reached on foot. In these cases, the driver will take you to the nearest point to the hotel and, if necessary, contact the hotel porter to assist you with your bags.
It takes 3 to 4 hours to get from Rome to the Amalfi Coast, depending upon traffic conditions.
Yes, there are a number of service areas along the route where you can stop for a coffee or to stretch your legs. There are also public restrooms.
It takes about 75 minutes to get from Naples to Positano by car, though the trip can take longer when there's heavy traffic. To get to Amalfi, it takes about 90 minutes and about an hour to get to Sorrento.