Can I convert the sharp edges of a polygon easily to round edges?

Try

polygon.buffer(10, join_style=1).buffer(-10.0, join_style=1)

A dilation, rounded (join_style=1), followed by an erosion, rounded. There's a diagram of a similar procedure at The Shapely User Manual.

For rounding your convex angles, you can apply successively a negative then a positive buffer of the same radius. For the angles shown on your figure, you are thus applying the method on the blue polygons.

The buffer tool in shapely is described here