love the chicken
okay--here is 'the problem' as I see it: the competition and the shadiness
any 'groundcover' that could out-compete grass and dandelions between these nice stones would have to grow thick enough and tall enough to smother the competition. I could suggest a couple of shade tolerant thugs which would also smother the stones and the walkway and you'd probably still have some grass and dandelions...
I am presuming that for both aesthetic reasons and functionality that you may prefer to have well behaved, short, and step-able cuties between the stones.
In my experience, stones never serve as a barrier to grasses but rather as an 'encourager' for the grasses to send runners and grow between. And likewise with weed seeds, they create wonderful, protected germination sites.
Personally, I hate landscape fabric/weed cloth and I don't use it now (although I wish I had in a situation that I have currently with a section of rock border--I did use it in the MW for all my stone paths because the weed problem there was
exponential ).
Anyway, you could go to all the trouble of removing the stones and base material and some native soil, lay a weed barrier and put in a solid border/barrier along the 'grass' side, and then replace it all and plant with things that don't need a deep root run. It makes it easy to pull weeds when they try to grow and would reduce the grass invasion tremendously. A ton of work to start but would cut the maintenance time. I could see your native moss as well as the scotch moss being quite content with that and maybe some violets
Cyclamen Leaved Violet (Viola 'Sylettas') -cute, sedum album, mini armeria, toadflax? I'm thinking more of clumping things rather than creeping things so that they don't eat your rocks...Semps do pretty well for me in shady areas--but don't like to be stepped on so much--maybe along the base of the rock wall?.
More creepy but still clumpy--cymbalaria, arenaria. And if you're more interested in creeping things that would cover the rocks, well there are some that like the shade (veronica, lysimachia--thug, leptinella, etc.)
I would suggest some type of barrier along the 'grass' side to give whatever you plant a chance because grass keeps being grass. ( I have had fair success just pounding a strip of glass-board along an edge like that. I use it too for collars around suckering shrubs and other) It could be that that and some vigilant weeding initially is all you really need to allow whatever you plant to establish.
For the expansion joints in more sun...thinking on it...
do you want something that stays more in the crack or something that eventually eats the sidewalk?
I have an ever-expanding antennaria that could literally carpet the sidewalk!
here it is eating rocks
I would be happy to send you some sheets of it