And couldn't you take out the equity in your house and make that investment anyway without losing the asset?

The biggest is that if you do not make the payments on the HELOC you will lose your home to foreclosure.

We lose lots of deals to the hedge fund companies.

The problem I have is that any variation of the word 'invest' makes my wife think that a coin flip will lose us our entire life savings.

From your spreadsheet it looks like you're losing money every month.  

You might want to do the learning before you do the losing and BiggerPockets.com/starthere is a great jumping off point.

Seasoned flippers or investors tend to put bids on properties like this knowing that 1 or 2 will lose money for every 10 deals they do. 

And now that I've learned more about real estate investing, I've realized that when I add the HOA ($430), property tax ($300), insurance ($50) and property management ($70), with rents around $1350, the best case scenario is a property that cash flows $500/mo after 20 more years of losing money on the property when the property is free and clear.  

Paid-off houses is just moving assets from higher risk (and thus higher reward, i.e. you have a leveraged house which frees up cash to buy another house, increasing your return but also adding some risk) to lower risk (if the place goes vacant for a year, you won't lose it to foreclosure, for example).

Seems like we would have at least a year or two of good rents before Nashville loses its cool factor (but who knows).