The song that Lucifer sings at the beginning is Sinner Man - an African American traditional spiritual song about a sinner attempting to hide from divine justice on Judgment Day.
Lucifer (Tom Ellis) sings a cover of "Sinnerman," most famously performed by Nina Simone on the 1965 album "Pastel Blues."
When Dr. Linda reminds Lucifer that before he fell he was known as "Samael," he replies: "I don't go by that name anymore." This is a quote from the original DC comic "Lucifer." However, in the comic this quote is referring to the name "Light Bringer" and not to "Samael," to which Lucifer still answers.
According to Tom Ellis in an interview with Valerie Morehouse (his vocal coach) for Lip Roll, originally Lucifer was only going to play the piano at Lux. But, during shooting early in the first season, he and some other members of the cast and crew went to a karaoke bar, where he sang a couple of songs. The producers were impressed enough with his singing to incorporate it into the show, with this episode being the first one he sings in as a result.