Where are you located, Ehab?
I am a Building Science Thermographer, and this is the work I do.
More information and testing is required, but at this point you should consider that there are only three sources of building moisture, building envelope leakage (air or water), plumbing, and condensation.
You seem to have excluded plumbing. Take a stab at the other two.
Construction practices, blueprints, etc will help with building envelope. What is really happening inside the wall. Fiber optics may be needed to see what you can’t see. Walls must be waterproof, but must be allowed to drain and ventilate to dry. But this is not foolproof either. After the big Nashville, Tn flood, I found several “flooded” houses where the source of water was the weep holes (improperly placed, and in excessive number required by code).
Moisture inside the wall should be determined. We really can not exclude capillary action going on here.
Condensation is always a fun one for me. After the process of elimination is done and excludes plumbing and wall leaks, I turn to condensation. It takes persuasion to get the facility engineer of the other two however…). Condensation requires specific conditions to occur, and these can be measured. Psycrometrics tell you what you need to know.
Note: HVAC design can play a large part in this (depending on where you’re at, which is why I asked). Air stratification can play a large role in this, as air distribution can put the condensation thing over the top. This can be an Air Leak turned water problem. Check building pressure. Air leakage can not occur w/o a pressure differential.
I noticed in your third pic that there is the same damage in adjacent rooms. You stated this is happening everywhere and on the 3rd floor. This problem has a wide scope, beyond an isolated air/water leak.
If this is just a Home Inspection, you may never know the answer…