Purposely let Obi Wan live? In the 2 encounters between the two Ben comes out on top. Once because he was simply better and the other time because he understood the nature of the Force better and sacrificed his body. Who knows what would have happened on the Death Star if both Vader and Ben were going all out. Although I would put my money on Obi Wan.
And Ben’s line “Your father wanted you to have this when you were old enough but your uncle wouldn’t allow it.” certainly seems to imply that Vader was aware of Amidala’s pregnancy. The fact that he didn’t know he had a daughter means he was not present at the birth. Was Ben lying about Anakin’s wishes or simply extrapolating from a comment made before any pregnancy? Sure it’s possible but I see no reason to assume it was so.
So I say he did know he was to become a father. And since nine months later there was a little Skywalker being raised by his brother Owen I can’t imagine him not fairly quickly being able to put 1 and 1 together.
So why didn’t he kill him? What for? I mean this was his own son and there was still good in Vader. And the odds that Luke might become well trained enough to take on the Vader or the Emperor individually seems extremely remote. Looking at the climax you see that for all the training Luke got he was still hopelessly weak against the pair. The only way he influenced the outcome of the final battle was by failing to turn as he fell to the ground helpless and begged for help.
I imagine that for schemeing minds like the Emperor’s he might have preferred to keep the lad alive on the possibility that he could be turned. Vader was strong in the force but broken in body. And if Luke did prove to be too much trouble kill the boy then.
Also I don’t think you can find perfect logic in any of the points in these movies. Lucas has many drawbacks and seems quite happy to fudge the facts when it suits him. And we might find out one way or another in a few years.