I've heard you can't stain pressure treated wood if its still "wet", as in the moisture content is high the recent pressure treatment.Wood isn't that cheap, but I still have to remove the old boards and install the new ones, speaking of time.

I would use roxel  insulation if any chance of moisture againHow do you know the wall left of this one isn’t same shape?

Assess ALL moisture invasion issues - Longer you wait the more expensive they get so those are a repair now thing.

Once the sun sets on a warm day you can also do a thermal scan, the wet areas will show up “hotter” in the scan because the moisture in the roof system will be heated compared to the cooler evening area.

Make sure you understand ANY moisture invasion issues. 

Adding a privacy fence to the property, HVAC system, and new moisture barrier.

The bigger reason I need this info is due to moisture coming over into my house via the party wall.

First off, get a knowledgable inspector to tell you what you're dealing with and suggested remedies (if needed).In general, moisture invasion is one of the bigger (i.e. more to fix) and more common items laying waste to property.WAITING TO FIX MOISTURE INVASION MAKES IT MORE EXPENSIVE>

This could be a function of the paneling and firring strips pulling moisture out of floor, or could just be a damp basement.  

I wouldnt, unless you have an eagle eye for moisture, foundational issues, cracks, roof condition etc. ~$500 now can save you thousands down the road as i found out last week!