I am no expert, but from the way you are phrasing your questions, it sounds to me like you still have plenty of work to do in the learning and research stage.

I just thought of a new phrase: "Controlled eagerness".

I would phrase is as "We saw many properties that date ( true), and we were surprised to realize a garage was not there after our second visit" We are interested, but will lower our bid to $X.Will the listing agent get in trouble for this?

It was an interesting negotiation process that got me to that point, but I followed the PB podcast tip that talked about never making an offer and always phrasing your responses to have the owner make you an offer. 

What was discouraging was that from how he phrased it, it sounded like every single foreclosure or bank-owned I might go after will want 2-day proof of funds... which was never discussed or available from the lenders I've spoken with so far.

This will straddle two tax years and that's another situation that will work to your advantage reporting wise and is a phrase that tax courts have used in the past to describe a valid exchange.  

If they follow with a "I cant" or even seconds before; If I could rescue them from this evil killing phrase I follow immediately with "raise your expectations" Theirs one thing you must be sure in business one pribciplw every group must have to have lasting durability "accountability".In 2005; or so.

I even posted some time ago for this very definition as the phrase is coined in BP.

How about we phrase it something like:"Healthy, green, environmentally equitable, socially just, cross-fit house for rent. 

The phrase "HELOC on my rental property..." is an oxymoron.