Keep your energy high and mindset positive.

Yes, you will have access to a VA loan with 100% financing, however:This doesn't to you a whole lot of good if you are only in a duty station for a few monthsDepending on your rank, duty station, and other factors, you may be required to live on base at timesVA loans are only for owner occupantsYour purchase price will still be dictated by your credit and income (see first bullet above), andIt can be extremely risky to buy with 100%+ financing at the peak of the market (I say 100%+ because many servicemembers and veterans roll their VA funding fee into their loan, which can leave them leveraged at 102% or more - with negative equity).

If you know you plan to sell it, bite the $500 bullet and pay a contractor. $500 is cheap in the REI game to make sure you aren't causing yourself larger headaches later down the road.

I'm not sure the electric bill would be higher, as central heat & air is usually more energy consumptive than a window unit. 

I get calls all of the time from agents who have buyers that want to "Just measure the curtains" or "Wanted to see the Basement again to see if a pool table will fit" or "want to count how many energy efficient light bulbs they need to buy" or "To see if their camper will fit in the garage"The answer is always the same "I'm sorry we are unable to accommodate that request due to everything going on at this time".

Or just bite the bullet on the first deal or two and build the relationship until you win the PM's confidence to invest alongside you.

This property is a major problem and a huge project, it is going to take lots of time and energy and stress.

If it were me, I'd probably just prepare with a sheet of notes with bullet points for each claim of damage and why I don't feel responsible for all/part of each item. 

That being said and to answer the question actually posed by your clients, there is no bullet proof protection for any asset. 

As you found, a rejected application because of your bad referral means you are stuck with the tenant.Eventually though they will only be able to rent to folks who don't screen, and those are usually the worst places who don't care about bullet holes and pests and rodents and code violations.Don't ever stop your eviction process until they are out of the property and you have keys in hand and have assessed all the damages.