Billy's is a bar that was owned by Will. Will would let Damon feed on people there and in return Damon would give him their ID's. Damon used this bar in the 1970's and was still when Damon revisited over 30 years later.
In the 1970's, Lexi was sent by Stefan to help Damon because his humanity was turned off. She had spent several months with him and every night she would talk to him about Katherine. Damon told her that he had feelings for her, and shortly afterwards they slept together. However, the next morning he had taken her daylight jewelery off and left her in the shade of a door.