I am very gentle and figure out there needs (why they are selling) then I create a plan that works for both of us.

Things to look for is a waterline on the walls an excessive amount of moisture damp floors walls bowing and cracking and the worst mould. 

Though it doesn't address repairing the foundation damage, it may give some insight into managing basement moisture in general:

"Moisture and dirt are entering the basement near the stairwell""Additional support columns/beams lack visible hardward securing them into place"Engineer noted there is no structural component dug below freezing level on any of these supports which would need to be done to get the foundation supports to code"Signs of water penetration noted on the basement walls"Picture showing proximity of nextdoor house"The soil slopes towards the foundation, or is flat, and must be corrected to prevent water from finding its way to the foundation" (This is the side of the house with the failing retention wall in the basement and has extremely limited access) The engineer recommended potentially just paving cement between the houses to seal it and recommended against a french drain as installing it would go below the level of the foundation already.  

I presume you have considered other common issues with basement units such as moisture, separate entrance, etc.

Carl,As a fellow property manager and real estate broker I think there are some key item to look for:Any moisture on the windows, around toilets, under sinks.

Two nights ago l gently tapped on their gate, can you please smoke further away its coming directly into our bedroom their yard is huge?

My 1st post so please be gentle.

Premium vinyl or laminated flooring.Laminated is a bit restricted to low moisture.

I haven't heard of anything quite like you're describing, but in general I always tell people to use a higher sheen whenever moisture is present.