I think Roger would be proud and disappointed in some of the Actions of both Arthur and Owyn

I think his reaction would be mixed. Lyons' chapter had the Outcasts splitting off from it because they believed Lyons had strayed too far from the Codex and the original mission of the Brotherhood. However, Roger Maxson intended the Brotherhood to be trusting of outsiders and ultimately be about helping humanity rebuild. So I think he would also be proud of the work Lyons is doing.

He would also be proud of his descendant Arthur Maxson for showing great leadership and reuniting the Outcasts with the East Coast chapter. But I think he would be disappointed with his genocide against synths.

Yeah, definitely mixed. If you listen to what Roger actually wanted the Brotherhood to be, the ECBoS and MWBoS are the chapters to come closest to his vision. The Brotherhood out saving people and collecting, even improving, technology to ensure civilization can eventually be rebuilt. But both failed in other aspects. In the ECBoS's case, Roger vehemently opposed the usage of nuclear weapons (even when his own followers were faced with a dire threat that nukes could easily finish off) and never specified that the Brotherhood solely save human civilization, things the ECs ignore.

I think Roger would most definitely approve of Lyons's rule. Roger wanted the Brotherhood to accept outsiders into its ranks and actively help the wasteland, something that his son sadly didn't quite share.

Arthur's rule would be a bit more mixed. Its hard to say what Roger would have thought about synths, imo, but I'm not sure he would have approved of Arthur ordering Danse's execution.