Cleaning Up A Shape

[Box]

You can Weld those edges before Pushpull and you won’t get those edges, or you can use the soften smooth tool.
As to the offset, it will start as a curve but after a certain point it cannot be a curve and will become a corner.

Simple Question? Eliminating Line Removes A Face

Hold Ctrl while clicking on the edge with the Eraser. The edge is required to create the faces on either side of it. Ctrl+Eraser will soften the edge so it isn’t visible but still does its job. Alternatively, you can triple click on the shape with Select, then right click on the selection and choose Soften/Smooth from the Context menu. This will soften all of those edges around the curved surface in one go.