Yes though apparently parking in summer is difficult to find in downtown Dubrovnik. An Uber or taxi for that reason might be a good idea, especially considering the parking garage is both expensive and a good 10 minute walk from the old part.

I'd personally say, yes get a car and drive around. A drive up to Split is also worthwhile (though I'd say stay up there too) - the road is very scenic.

The Sheraton is half-way between the airport and Dubrovnik, it's modern and nice and in a new neighbourhood, but not much to see and do there except the hotel itself. A car will therefore be useful.