This is a theory that I think may be possible about what the connection between Liz and Red is. Ican't find any flaw with it. If you do, please comment to let me know.
Firstly, I don't want to sound dismissive but in my opinion, Lizzington doesn't seem possible. At all. What we have seen on the show of Red's feelings are those of a guardian not of a lover. When addressing the parallels between The Silence of the Lambs and The Blacklist, James Spader himself said in an interview that the Hannibal-Clarice relationship was based more on mutual attraction towards the mstery between them while in The Blacklist, Red seems to have a certain knowledge about Liz. Things she doesn't know about herself. About her past and her personality.
That being said, from what I've seen, I think Reddington was the lover of Katarina Rostova. It would be an added dimension if it is found out later that Liz shot her father when her parents were fighting over that very issue. It explains why he has cared for Liz so much in the past. It also explains why he cannot let go. Why he has to know what she's doing at any point of time so that she's never left unguarded. It explains the expression on his face when he sees the blonde Liz. I'm going out on a limb here but maybe his desertion of the US Navy was due to him meeting Katarina on some operation in Russia and heralded the beginning of a criminal career.
Moving on to the fire, I'm not sure which episode it was because I've watched every episode only once but I distinctly remember that in one, when Red removes his shirt, there is what looks like a large burn scar on his left shoulder and back. I don't understand how that has been ignored. I haven't heard it being mentioned anywhere on the Internet. I think he pulled her out of the fire that night. After the attempted memory extraction takes place in the episode 'Luther Braxton: Conclusion', Dr. Selma Orchard says that the roles of the people she saw in the fire might be muddled up. Of course. When Red is asked by Liz if he was there that night looking for the Fulcrum, he says, "It's not that simple." My take on that is that yes, he came for the Fulcrum but he also came to save her. Her father did die in that fire. So Red wasn't lying when he said that he wasn't the father.
I don't know how much of this has been thought of before but here it is... Let me know your views.