Like WasntMe, I think the childhood fire (and Liz's scar) is an important clue. Red may have been instrumental as a cause of the fire. If Liz was the only survivor, Red (in his twisted honor-among-thieves morality) may feel responsible for Liz's well-being. They aren't father and daughter although he clearly is paternal toward her.
The writers need to speed this "thing" up, however.