Yes if you can post some pics that would help with better responses. I've got some steep areas also.

I also live on a slope and have found that making terraces has been the only solution for making a garden. Not having a dog I don't know about dog runs. All our animals are natives and manage the terrain a little TOO good. Al

Terraces are the best solution, but very expensive.

I don't think you will find any material that will stick to weed cloth on a steep slope. I think you will want to plant some tough shrubs or groundcover to cover the weed cloth instead.

My father-in-law put two railroad tie winding staircases up his slope so the dogs could run up one, across the back, and down the other. He planted rosemary on the slope, with a few trees. It works beautifully. They prefer the stairs to the slope.

Renee

Here are the links to photos of the backyard. I don't think they do justice to show the angle. The slope begins at the cement barrier.

http://vijpersonal.zenfolio.com/p654278809/e28565fd2

http://vijpersonal.zenfolio.com/p654278809/e38226918

Your pic with the side fence does help in showing how steep. It looks to be about 8 feet or so wide or surface area. You might be able to get 1 terrace layer.
I also think a combo of steps and terraces would work.
You need to consider how big the dogs are and will they be running and trampling over everything...is it their playground.
Since I have no animals either I don't know what to recommend for plants. For dogs running around anything loose is going to get kicked up...wood chips, pebbles, rocks, gravel etc

Here is a pic of mine after I tore out most of the dead plants and before I started. I have since added a row of blocks to each and replaced the old rotting railroad ties with treated wood.

{{gwi:527042}}

It doesn't seem that steep. I'd build a small block retaining wall to flatten the whole area out and give yourself more usable space, which adds value to the property in the long term.

But, I like to do stuff like that, and not everyone does.

You'll regret that weed fabric pretty quick. Weeds simply grow right on top of it.