I am planning to buy properties in East Tennessee, South Carolina and North Georgia because that is where we want to put down roots in a few years.

I also don't know if there were signs that there was already a problem from the tree roots when the neighbor purchased the property.

You can build stronger relationship if you take the time to really grow roots in the market that you're looking to invest in.

I agree with what @Julien Jeannot said, REI meetups are a great place to start and get rooted in the landscape of REI.

They understand their liabilities and have a warranty policy, insurance, are ethical, and prefer to not "patch" things (preferring to repair the root cause, thus reducing call backs). 

In the backyard, theres a short tree with roots that visibly reach out distances of 10-12 feet or so with the visible ends under the porch.

Factors:age of propertyage of treetype of tree (a redwood has surface roots and spreads out massively, while others have a 'tap' root descending ~1/3 the height of the visible portion of the tree

A friend of mine is a landscaper , he had to replace a driveway for a guy due to neighbors tree roots ( neighbors dont get along ) Part of the job was root pruning , he basically dug up all the roots that were over the property line , ( BIG roots) and they tripped all the branches hanging over the property line .

I tried fertilizing it with muriatic acid, even cut the roots so it could absorb the fertilizer better.

A WA/OR lawyer, Darrin Class, has provided these helpful (for me) summaries of WA law on matters of trees, roots, branches and "Spite Fences" (like painting trees blue).