In the first episode he was barely hurt by any of those bullets(Delilah's Ex shot him with about 4-5 bullets) but in subsequent episodes its shown that lucifer isn't immortal as long as he is close to her. Any rhoughts about this inconsistency?
Most likely, the writers had the idea with Lucifers vulnerability-problems after the Pilot-Episode was released. Maybe they put this in because they wanted to make Lucifer more relatable. The show wouldn't be as fun, if the main character was absolutely all-powerfull. The Pilot-Episode is mostly for the writers, so they can test the premise of the show and react to suggestions of the audience. There are almost always changes after the first episode.
If you want to apply continuity-logic, it may be that he had to be near Cloe for some time, before the vulnerability was put in effect. Either that, or it isn't that his procimity towards her makes him vulnerable, but it is a psychological thing, like when Amenadiel lost his powers. Maybe his feelings towards Cloe make him vulnerable when he is close to her. That would mean, that he simply didn't love Cloe yet in the first Episode and his feelings developed over time, making him more vulnerable.