Lucifer, Stay. Good Devil. - S1-E2
Stupidity: It takes an awful lot of pointless visits and oblique thinking to simply investigate the known (and apparently only) collaborator and protege of the suspect, who also was on the crime scene. You'd think interrogating the business associate would be standard procedure.
Answer: As Cain theorises in a later episode, it's Chloe's love, not presence. This early on, Chloe had no strong feelings for Lucifer, so had no effect on his immortality. In fact even more recently he's back to being invulnerable around her after speculation that he chose to let himself be vulnerable. So while there's a degree of flexibility / retconning going on, it's all explained in-show.
Jon Sandys ★