"Everything else is adjustment to basis"

Not the mortgage payoff; but yes to the improvements as well as original purchase.

"Your question is whether the "gift of equity" is included in the selling price."

Ooh, mind-reading...

Honestly, it would help us to have some details @kabbabkkg regarding the amount of that gift of equity. Are you aware of the Gift rules?

Example:

https://www.investopedia.com/terms/g/gift_of_equity.asp