Basement water intrusion and dampness is like kryptonite to me.

The age of the property would put it in a safe harbor, however of note you would need an "intrusive" or sometimes called "forensic" addendum to the needs assessment (PCNA) that reviews major building components like the breakers and sewer lines for adequacy. 

This looks more like a combination poor install, flashing (if it is step flashing) definitely not overlapping enough to prevent water intrusion, and heat. 

Gutters could be the reason for some intrusion but might be more to it.

After insisting on it they did and it revealed they had to repair water intrusion into the basement. 

The majority of the sound wave intrusion in your room probably comes in through the window, so if you have that much outside noise I would start at the window.

The crawl has had water intrusion previously, so something would have to be done.Excavating the crawl and having it waterproofed from the inside is expensive and difficult.

First thing right away though is to fix up those downspouts and make sure any water intrusion stops asap.

If it hasn't ever been painted/coated since original application just doing that will help hold it together and prevent dust;  that work should not be too intrusive.

Also need to determine the source of the water intrusion.