Apart from the above mentioned, there are not many other things you could do, yet there are several malls and beaches you could visit. One of the malls that tops the 'all time favorite' list is Villagio mall. And one of the highly recommended beaches would be Khor Al Udeid Beach, if you are visiting Sand dunes, you could club this with it, Otherwise it could be a little difficult to get there. So, this is it. This list sums up most things you could do in Qatar.
One of the best places to stay is within Souq waqif or in and around it. It's where public transport such as hop-on hop-off buses stop and most touristy things are in walkable distance. Easiest way is to figure out using Booking.com's hotel map that helps you compare all the prices of nearby properties.