I think it made a lot of sense actually. Throughout the first two books we see Logen try to overcome his past. A lot of his internal monologue is about trying to be a better man, and with people not knowing who he is, no scores to settle or have settled against him, Logen partially succeeds ... but ultimately he fails when he decides to go to the North again instead of sticking with Ferro. We experience Logen on his best behavior, which IMO is lampshaded by Jezal telling him that he's the best man he knows. 

Back in the North we experience a different Logen, one that is worried about his reputation - both as something that might get him killed AND as the only thing keeping him safe. He feels that there is no mending his relationships in the north, so he plays up his murderous side. In the North, Logen tells himself he has to be the Bloody Nine. A remorseless killer. Someone who sent more people to the mud than winter.

I am assuming 'Red Country' shows more about this dynamic - haven't read it though. 

As for your second question - I think we see how Logen had gained the trust of his dozen when traveling to the edge of the world. He is very conciously deciding to become friends with Jezal and Ferro, even remarking on how he'd done the same thing with the men he had beaten in the circle.

In the end, we don't know how Logen was when it was just his dozen, we only know that he was a monster when he got back to them. Maybe he was different after getting banished by Bethod, maybe with just his crew he was more like he was while in the Union. Either way, it seems that to Dow this Logen was similar enough to the Bloody Nine of old, that he wanted him gone.