Who is responsible for coverage for excess damage to the property?

Is there a simple way I can cover the chimney up so rain/moisture doesn't get in but also allow proper ventilation?

If it's moisture creeping in from the ground, consider adding a dehumidifier or figuring out how to seal the basement.

Generally any excess cash I have is a result of refinancing or at the very least could be used to pay down debt, therefore it is part of the finance of the deal  

For most people the best formula is to create income (preferably multiple streams) elsewhere (job, business, both) and funnel as much of the excess income as possible into professionally managed income property.For example my day jobs (property management co and syndication business) create value and cash flow for me which I in turn funnel back into the biz and income property.You need big funnels of income to get truly wealthy (financially free) in real estate.

It can become rather complicated, but simple having caps where increases are carried forward that earns a rate of interest and adjusts later where actual payments may go up to a point and excess amounts while not payable are still assessed, accrue interest until the next rent amount adjusts then amortize the deferred amount over the next term together with any increase.In numbers, say a cap rent was 500, first adjustment is 70 more, but you have a 50 cap.

Also, people with sufficient funds will leave these areas and the area will go into decline.Climate - Properties in areas with hard freezes and excessive moisture will tend to require more maintenance than in milder and dryer climates.Age of the property - The older the property, the more maintenance it will require, unless all the major systems and components have been updated recently.Property cost - There is a tendency for new investors to choose locations with very inexpensive properties.

Would we need to do a test inside the wall to look for moisture? 

What pattern are you using, does underlayment need to be put down, do I need a moisture barrier etc. 

The property needs some foundation/moisture mitigation work, a trash out, and a full rehab.