How to design a "dome" featuring two curvatures in a solid part?

You could also use Surfaces if you need to have greater tangent and curvature control for the dome. To do this draw the boundary sketch(do not use Slot for this) and two constraint curves. Then use the Filled Surface to complete the shape.

You can do this with a loft. It uses a sketch around the edge of you base part, a point to loft up to, and an arc through that point. Loft between the point and your base sketch and then use the arc for a boundary curve. I'm sure there are others, but this came to me first. Hope it helps.