As an Engineer, I would suggest a lot of the problems may be coming from that embankment at the back.
Your description of what is happening suggest water is pushing the wall over, along with the solid just getting damp and loading the wall.
Many small walls years ago were built with nothing to stop them leaning over with time, caused by the weight behind it, even low walls, 2 feet.
I would have a good look at shifting that soil back, sloping the ground to remove the need fot a wall if possible, and improving the drainage so any water can swing past the hues.
If its not possible to dig back because of boundaries etc, remove the wall and build something designed for the job.
There are kits available that would required galvanised posts inserted into holes 2 feet deep for a 2foot wall and concreted in. Then concrete slabs, or thick timber slabs of suitable wood installed behind the posts.
AS you back fill the wall, start with screenings and a slotted pipe, so water pressure cannot build up.
That pipe needs to drain around the house again.
The screens column
should be about 5 inches wide as the soil is backfilled and if you can place a mat like shade cloth as well, between the soil and the fine 1/4 inch stones to prevent soil washing into and blocking the stones,
thats another better outcome.