You might change your strategies as you go through the motions of doing so my best advice is to just start. 

Sometimes putting up cameras outside and motion activated flood lights encourage people to move it to their back yards if thy have one.

- use moisture meter- use commercial dehumidifier- most likely the humid level would be reduced by dehumidifier, indicating there's actual waer leak- what you need to check here is the plumbing first, fixing stucco would not solve plumbing issue if there's water leak insidebtw if you are sensitive to allergy you would know if there is water leak or not lol you could just feel it

These questions need to be answered, but they will not give up that information willingly, therefore the need for legal intervention with search and discovery motions.

We also have motion/vibration sensors, moisture/temperature sensors, light sensors, etc.

Sending them an email at 10pm at night expecting to make a major change to the plan of attack that's already in motion?

If you are water tight they will just keep filing legal motions that force you to run up huge legal bills that will eventually force you to settle just to keep your sanity.  

The properties will be under various stages of construction, will have electricity on, but will not have traditional internet access.I am looking for something where I could possibly either use a portable wifi hotspot or 3G streaming to a cloud-based DVR type setup that I could access any time using the internet, and would have large enough storage to hold (at least) a week's worth of video, and would have night vision and motion sensing technology, and support multiple cameras for different sides of the house.From the research I've done so far, it seems like most wireless cameras like this require a static IP address (i.e. a portable hotspot would not work). 

In Washington one needs to make a motion with the court.

No guarantees, always risks but if you get into the motion it is the way to go.