I find it strange that Vito couldn't bring himself to participate in Carlo's death - did he atleast agree that Carlo should be killed but just did not want to be the one that has to make that decision ultimately?
Basically. The novel makes it clear that pretty much everyone (Tom, Michael, Vito and Mama Corleone, likely Connie) knew of Carlo's treachery but pretended not to know. This is why Connie asked Michael to be Godfather to her child and in book was constantly seeking (via Kay) some affirmation from Michael that Carlo was doing a good job and well-liked.
In book Vito never speaks about Carlo's planned demise openly but Michael says everything is in place; it's the right decision and that he takes all responsibility. Vito says something along the lines of "good b/c there's some things even he can't do." Tom later confirms to Kay that Vito was in an impossible position. In order to avenge his son he had to murder his son-in-law. Even though Vito approved, by Michael taking responsibility (helped immensely by Vito's death) Michael took the brunt of Connie's rage, spite and grief.
Per book Connie was arguably his favorite child so Vito didn't want to hurt her like that.
Carlo's murder had to be simultaneous with or after the murders of the other victims otherwise it would have tipped them off. So Vito and Michael had to wait.
I don't think Vito ever would have countenanced or advised the murder of his children. Of course if Vito were still around Fredo would not have had the resentments he had nor would Roth have thought he could get to the Corleones via Fredo. I do think that Vito might have been a bit more solicitous or even aware of Fredo's feelings and have found an oblique method of dealing with them.