With that said I also find it confusing, as one street can be nice but the next two or three streets over could be another story.

If you cannot get a sufficient appraisal or qualify for the re-fi, you would be foreclosed on...the seller may ever insert that you would quit claim the property back...there are a lot of moving parts on this...

Most probate properties are not the kind of deals with sufficient margins for a quick turn.

I've helped three clients find duplexes in the past 4 months, so they are out there.  

All three units are currently occupied. 

I looked at three today myself

I have three more days of due diligence on my first investment property which I am planning to either flip or fix up a bit and rent out.  

All three units are currently occupied.Ok, I am interested in moving forward with the process of potentially purchasing this triplex: In what order do I proceed from here?

A relative may be sufficiently motivated to work with you.It may be that a formal Conservatorship has been established.

You would still have the two requirements of 1.purchasing at least as much as you sell (the net sale or roughly 700K) 2. use all of the proceeds in the next purchase (roughly 175K).If you keep that property as  a rental for no more than three years after living in it for two years you would be able to sell it and still qualify for the primary residence tax free exclusion even though it is a rental at the time of sale.