Both are correct. "Go over the roundabout" means that you're crossing a junction where there happens to be a roundabout. This means that you are continuing in the same direction. "Go around the roundabout" ( or more commonly 'round' in B.E.) means that you are circling the roundabout. In the UK, and Ireland (and any other country that drives on the left and has roundabouts), this would mean that you enter, follow the flow of traffic round to the right, and exit the roundabout on your right. "Go around the roundabout" could also be a means of turning back on yourself. If you realise that you're driving in the wrong direction, you could go round the roundabout a full 360 degrees and then drive back the way you came.

I think that depends. Normally we go round the roundabout or turn at the roundabout, but someone could go over the roundabout when he's drunk. in addtion, I'd go with "go over" if you're talking about something like drones.