Fence and wall have been leaning like this for about 10-15 years
... so a stable condition. Does the neighbor now want to buy a new car so that your leaning Pisa fence won't fall on it?
You wrote (at least over there) that the neighbor's construction is already twenty years old; then the structure must have well withstood the neighboring terrain depression beforehand. The "guiltiest" factor seems to me to be the pressing backfill on your side. The car in your planned carport seems to me to be the greatest threat here, and will trigger even more landslide in its direction.
what would be the most cost-effective option, setting L stones or a retaining wall made of formwork stones?
Formwork stones are cheaper. But you will probably still have to remove some rows of stones from the driveway to properly construct the foundation. It’s not done "just like that" even with formwork stones (reinforcement must be included as well).
what would be the most cost-effective option, setting L stones or a retaining wall made of formwork stones?
Summary: both are equally unsuitable, you are oversimplifying it.
Then I will probably have to have it done. Can someone estimate the effort, how many steps would that be? How many working days?
Whoever does it doesn't make the difference. You cannot fix poor planning by commissioning a professional company to execute it. Including planning, about one week in man-days.
