To hide the edges between faces that lay on the same plane, use a Decimate modifier above your Wireframe modifier.
Set it to Planar mode, and adjust the angle limit if needed.
The inconsistent line thickness is caused by the original mesh having been scaled up in the three dimensions with different scale factors. The mesh "remembers" these factors and applies them to the modifier results, making the wireframe stretched weirdly. To fix the problem, you must make the mesh "forget" that it has been stretched, and make the new scale permanent ("Apply" the transformation). To do so:
- select the object
- do Ctrl+A > Scale.