A mesh that’s part triangles and part quads will always produce problems when subdivided due to the fact that these types of polys are handled differently. Quads are divided into four quads, and the surface is smoothed. Tris are divided into three quads, and the surface is not smoothed. So if you have both types of polys in your mesh, the model will be partly smoothed and partly not. This is why it was suggested that you turn Tool>Geometry>Smt off when you divide the first time; it allows the entire mesh to smooth uniformly when you then turn it back on for the second and subsequent divisions.
The puckers, though, are due to points where more than four polygons come together. These “stars” always produce a pucker and should be avoided.