What is the connectivity between on-center & off-center bipolar cells?
First off, the post is quite lengthy and contains various sub-questions. I will answer the first two questions in detail, but only after starting with an abstracted overview of photoreception in the retina. By the looks of it, your basic understanding of photoreceptors and secondary visual neurons is incorrect. Clearing the general neurophysiological workings of the retina up should get you on your way in understanding questions 3 - 6. 

Synopsis of information flow in the retina: sensation of light
In Fig. 1 a schematic of the retina is shown. It basically consists of three layers. Layer one contains the photoreceptors, namely the rods and the cones. These primary sensory neurons absorb the light with photopigments and generate electrical signals in response to that. The photoreceptors, in turn connect to the second layer, namely bipolar cels, horizontal cells and amacrine cells. These cells are responsible for the center-surround structure of the receptive field of the last layer of cells (Fig. 2), namely the retinal ganglion cells in the front of the retina, whose axons pass across the surface of the retina and form the optic nerve (Kolb, 2017).  


Fig. 1. Retina. source: Quasar Group  

Note that there are no light-sensitive cells in the retina besides the rods and cones (barred the retinal melanopsin ganglion cells, but please forget that for the sake of simplicity). This one detail should basically clarify the sub-questions 3 - 6 in your post. The first two questions I will address next.


Fig. 2. ON and OFF center responses. source: McGill University

Regarding the retinal bipolar system and your first two questions: 


  
  Do rod-photoreceptors only form synapses with on-center bipolar cells (and on-center ganglion cells in the next step)?  
  and  
  Why do rod-cells only connect to on-center cells, while the cones connect to both on- and off?
  


Rod bipolar cells in the mammalian retina are of the ON type. In contrast to the cone system, rod bipolar cell signals are split into ON and OFF components by amacrine cells. The cone system piggybacks then on the cone system; Rod ON signals flow from rod bipolars to amacrine cells to ON cone bipolar cells. Rod OFF signals are generated between amacrine cells and OFF cone bipolar cells (Nelson &amp; Connaughton, 2017). See Fig. 3 for an overview of the retinal connectivity between rod and cone bipolar cells.


Fig. 3. Retinal circuitry. source: Nelson &amp; Connaughton (2017)

References
- Gouras, Color Vision: In: Webvision: The Organization of the Retina and Visual System (2017)
- Kolb, Simple Anatomy of the Retina. n: Webvision: The Organization of the Retina and Visual System (2017)
- Nelson &amp; Connaughton, Bipolar Cell Pathways. In: Webvision: The Organization of the Retina and Visual System (2017)