What is the best thing to do with the side of my house that has no sun

It would be in your best interest to put concrete in for the dogs

There is a section like that at my house. I put a tumbled gravel down with stepping stones and added yard art for interest and also have a couple bird feeders too.

Stone walkway from gate to other part of yard. Perrenial vines on fence with a couple mirrors hung between them (positioned to reflect bench). Plant Impatiens or Begonias (both shade flowers) along bottom of fence. Hostas at foundation. A bench and small table placed against the house facing fence. Tada! Secret, ,tranquil relaxing spot.

I would use stepping stones in two colors to make a pattern...eg. checkerboard and edgeit with pea gravel. I can then either purchase the checker pieces or make them (use dense foam for seating and cover in outdoor fabric) so I have a built in fun game for family and friends. In the event the dogs use it for their washroom it all washes off and drains into the pea gravel.

Looks like you have sun . Why not build a pergola. It would make use of an already private outdoor venue. Or make it into an outdoor kitchen.

Doesn't the North side usually get that green mold? Everyone in our area gets it. I was told to go to Lowes for a special cleaner.

our south side has no sun. I have a plot of soil there where an Autumn Olive bush thrives(but it can become invasive throughout yard, so be careful!), plus bleeding hearts grow there, creeping flocks, bee balm, Sweet Williams. Be sureto keep any plantings at 18" or more from foundation to keep bugs/ants at bay. Gram E

I do crushed stone/pebbles and a small area of mulch.

I would put pea gravel on the whole thing, with stepping stones in middle. Then you can put assorted potted plants there, there is plenty plants that like no sun. Maybe add a bench, or couple of chairs. You could gather some stepping stones up against the wall in a small area, for chairs.

Put shade plants