Painting over wallpaper is a half measure approach and just leaves a problem for someone else.

Should we approach this differently?

If you use an FHA loan for this approach, the down payment will be just 3.5%. 

I’d like to approach landlords to see if they want to sell me their property in exchange for a $10000 down payment, $5000 of which is used for closing costs, and $5000 of which is paid for with seller credits.

I, personally, take the conservative approach and budget for a roof today, over the remaining useful life.  

We are well past the "Break Even" occupancy rate, covering all costs and Principal/Interest, thus this is not a low "Cash Flow" issue, but a Rent Up approach. 10x20- about 1/3 full. 

I'd also appreciate thoughts on whether short term or long term investment properties are a better approach (or one of each) for someone in my situation and near military?  

The stage is now set, based on your approach, and will lead to a certain dynamic in the negotiation

That's not my approach, personally--I'd rather just negotiate a great seller financing loan for as long as possible, and avoid the headache and hassle of refinancing for as long as possible. 

Any security deposits and the last month's rent that the tenants paid to the previous landlord should have transferred to you at closing.From a tenant happiness perspective, it is worthwhile using the approach of showing them the rental comps and then asking what they think is fair for an increase if they are considerably below market.