My initial choice was to buy here in CO using the cash as 30-40% down payment, but then I got nervous I'd be buying too high.

Not sure of how my mother's eventual passing, will change that, but I anticipate that he might feel that her passing eliminates the need to generate income from that unit, and he can resume a rent free status.

My strategy has been this: I will initially pay cash for a house in this price range.

The invoice include all kind of small things he had listed in his initial demand after the inspection.

There are many options and taking cash from one of the buildings that is generating income is definitely an option provided it is not in an LLC, Corp. or any other structure that would make this co-mingling of funds. 

My initial deal, a 3-flat, cashflows about $800-$900/month.

I wanted to know if I could pull a HELOC to invest that money into purchasing another property where I could generate passive income.

Or do improvements not even matter that much since most of the appraisal value will be based on the income the property generates? 

Very minor TLC needs to be done from the looks of it, but I do think that some improvements to the floor plan need to be done initially (I am in a house hacking situation where I need 2 bathrooms, and the house only has 1).

I was initially thinking, if this is a complete gut job, everything will be essentially be brand new and i maybe able to attract decent tenants.