First of all, my hexagon would not fill in until I drew 2 lines across it.
Then, when I tried using Push/Pull to make it thicker, the lines around and within the shape got thicker. However, the face within those lines moved without getting thicker. (See the attached file.)
Hard to say exactly without seeing the model but generally it’s best to not divide the face into smaller faces before extruding. Looks to me as if Push/Pull was doing exactly what it is designed to do. It might be that you could have pressed Option before starting the Push/Pull operation to generate the faces.
To help achieve your goals, some time spent at the SketchUp Campus and at the SketchUp - YouTube channel will be very worthwhile. Both sites are from the SketchUp team. On the YouTube channel, pay attention to the Square One Series and Level UP series. They cover the basics for each tool.
Yep. The OP might not have realized that means the vertices of the shape are not co-planar, hence can’t create a face, hence can’t be pushpulled as a unit.