House was for sale on court house auction. bidding went to $59.900 . bank bid for 62K. and was offered to bidder.  

There is no almost-married, split property in half fairly court:)

It may depend on court case precedents in your state. 

Moreover, the court that enters the order for tax deed will also enter the order for possession so there is no need to go to eviction court (called forcible entry & detainer court in IL).

For ex. suppose you were to loan someone money say 10k and either took out a mortgage on their home or used something else of substantial value well in excess of the loan as collateral, how would the excess be handled in court or otherwise?

What is the difference between the court appraised value vs. finding out comparable sales, ARV, zillow, redfin, Cash sales around the area?

If they don't, do they have any basis for obtaining it via courts or other means? 

One county sends out a summons to court.  

We really hashed through it all and got very nitty gritty.We have no idea if our document will actually hold up in court, however it does document our full intent to everything and when we signed it, we both agreed to honor it.  

The full HOA debt is yours and the HOA has little incentive to negotiate since they know you now own it free and clear, except for their debt, so if they foreclose they know they'll be paid in full, plus any attorney/court costs.