1. 3) Use modify/mesh to smooth the interior nodes
3) Use modify/mesh to smooth the interior nodes. Edge nodes will not be moved. You can hole individual internal nodes also if desired by selecting them as hard points on the modify/mesh form
1) Starting mesh with new edges in yellow
2) Project edge nodes onto desired curves
Modify/node/project. I had to move the lower right corner node using modify/node/move.
Note: while the edge nodes can be moved off the curvature of the surface, the internal nodes will stick to the surface and not be smoothed outside of it. You can disassociate the mesh from the surface and re-associate them to any surface that may be big enough for the smoothing to do what you want.