In this context, "Come on" is a gentle suggestion that the person crying should come over to the singer, to be comforted I think.
"When the night falls on you" is not commonly heard, more the language of poets, as is the next phrase, "Take me into your darkest hour". Very descriptive, and leaves one with the feeling that the two persons concerne are very close.
Well, it's a hard question to answer. Most of the songs I've been questioning are in my mind. I retrieve them from my brain and try to understand them a little bit better. I hope learning English can be great fun through listening to music, especially the ones that I truly love. I might not be on the right track, but still enjoy the challenges very much. I shall move on, thanks to you, folks.