Taking an interest in his day seems like a good start in warming up towards him, FS. After all, if you are ever going to R, at some stage you will both need to open up about your lives to some extent and start sharing things more. Baby steps, I know you're like me and a distancer so it doesn't come naturally, but if you keep it light and low pressure, good things may flow that way.
Dh and I have been definitely showing more interest in each other's lives the last few months, it's only now I realise how disconnected we really were before because we didn't really show interest in the other person, we sort of paid attention out of the corner of our eyes I think. Although it's not enough for R, it has to be a first step towards it. It comes easier once you decide to trust your H a bit and to not be too guarded. I'm working on that. Yesterday dh said something about 'my road' being closed (the one his flat is on) and part of me tensed up and then the more generous part of me said 'chill out' and we moved on. Trusting someone who has broken your heart is probably the hardest thing in the world though.