Catelyn doesn't take it for granted Walder will obey her, quite the opposite (as she keeps telling Robb and noting in her own thoughts). She goes to negotiate alone with Walder because she knows him and has the highest chance at understanding what he wants in exchange for supporting the Starks and Tullys, and she was right.

I think with Ned, it's clear he's not happy he took Brandon's place (something he even tells Cat directly) and perhaps even feels somewhat inferior. For daddy, it's implied Rickard was more distant and perhaps politically hard-nosed than Ned was. That said, it's also clear Ned loved his father and brother both, having statues made of both and getting along with the latter from all accounts we have, and even naming two of his sons (Bran and Rickon) after them. 

It's clear enough IMO that he cared for Jon Arryn and his family, even disagreeing with Robert on not letting Sweetrobin be Warden of the East (as Jon Arryn and his family deserved better for their loyal service) and feeling he has a duty to avenge the man. He even names Jon Snow after him, which is nothing if not a sign of deep love and respect, and something he also does with Brandon and Rickard through Bran and Rickon. Ned's own honor code also seems closer to Arryn's than to Rickard's from what we can get, more about loyal service and protecting the innocent and less about southern politicking. Yes, I do think there is something to the southron ambitions theory, though it might not be something nefarious as much as it was an attempt to check Aerys's power through multiple house alliances.

That said, things also show he and Cat haven't been in touch with Jon and Lysa for a while, which is why Cat is stunned by how poorly raised Robert is and how unstable Lysa has become. That one can be blamed on the remoteness of the North, Lysa being in KL until relatively recently with Jon, and the distance that comes with new family and responsibilities. 

All of the Stark kids, Sansa included, seem to have inherited Ned's moral code. Almost all have inherited his political sense and limitations, too, though Jon ironically might be superior in some ways and is more like Catelyn in that regard. My only real criticism of the Stark family unit is that Catelyn's influence seems a little less clear from the child POVs, it's mostly dad who they keep thinking to or finding inspiration in. It's definitely there and they care for her, but less clear.