Tywin's blatant hint-dropping was on purpose. He wasn't gloating or even threatening Ned - it was supposed to come off as a warning to him. Something akin to 'watch out 'cause someone's going to be shanking your lady soon'. But Ned is a tad busy with matters of the crown and thus didn't provide Cat and Robb with adequate protection. At least, that's what he thinks.
1. You might want to rewrite that bit then. Because I don't think anyone on the site took that as a warning. And it coming from Tywin...
2. How busy can Ned be? We've not seen or heard mention of a small council, Jon Arryn, a continuation of the war, anything besides the coronation. Which can easily be delegated until... you know, the new royal family is present.
Doesn't mean Tywin actually wanted Cat to live, of course. He wanted to come off as sympathetic, as someone who'd tried to help Ned.
As I've said before, there is nothing in your set up leading up to this point that would indicate Ned would even see Tywin in that respect. It's like me suddenly appreciating a mugger who pops up and tells me that he warned me I'd get mugged.
As for The Baratheon brothers... well, I never got the impression that Robert was going to be crowned from the beginning of the rebellion. Maybe I'm wrong, but the Baratheons sort of lost all claim to the throne when Robert died. I mean, who'd follow two guys with neither the prowess or charm of Robert?
That's... that's not how inheritance works. Robert had the strongest claim because he was the most visible warlord in the rebellion and the one's with the closest blood ties (which meant Jon Arryn could build something of a coalition behind him for the post-war). The Stormlander lords might rally behind Stannis on principle, regardless or not of their respect for Ned. Ned himself would push for Stannis as rightful heir, and that needs some coverage as to why he didn't. And why the Baratheon's were pass over. This is the stuff of Great Councils in ASOIAF canon. It's why Aerys's branch of the Targaryens became king instead of the Night's Watch Maester or any of the youths or heirs that had been born from guys like Brightflame. It's suspected to be part of what Rhaegar was planning in regards to his far after the war.
IDK, this made sense to me at the time.
Cat's brutal death was going to be a plot point for me later on. After Ned is crowned and marries Cersei, he was going to lead an intensive investigation into Cat's death to find the killer. This would've led to things like character development, political wrangling, and so forth.
All the points you've brought are perfectly reasonable and I hope that I have, at least, made clear to you why I chose to go down this route.
So... just to be clear: He's too busy with his coronation to provide a proper guard for his royal wife and son (ignoring the fact that Hoster Tully would have raised the host of the Riverlands for his daughter the minute she crosses into Frey lands, or that she could have come by ship around the Narrow Sea, or that Jon Arryn would have happily handled this sort of thing for Ned), but he's going to be more than free as King to investigate his former wife's death, and despite being king... is going to be involved in political wrangling over that.
Not to mention how even in your own writing you seem to be rather ... rushed (?) to head past Catelyn and Robb's deaths. Noble lords throw their children at Ned (who are these noble lords? Rebels would know he's married, loyalist must be drinking shade of the evening if they think they've got anything resembling a chance of being in the new royal court), Tywin comes across rather blatantly (granted it might because we know Tywin, but besides him, whose got the resources and motivations to do something like this? The Targaryens crumbled rather fast in the wake of the Rebellion as an organized party), and even Ned himself goes from (oh god why) to (well, I'll have to do better by my next wife). How Ned like is that?
ASOIAF is a series so delicately balanced that doing one thing at the right time changes everything. What if Lyanna lived, and Robert took her as his queen? What if Stannis failed at defending Storm's End, and the Tyrell's took the Baratheon stronghold? What if Aerys set fire to King's Landing as Tywin's army sacked it? What if the Martells decided to keep fighting after King's Landing was sacked? What if Robert died at the Trident or soon afterwards, leaving Ned to pick up the pieces?
In here, Ned's a particular character in a particular setting, surrounded by people who's history with him hasn't very much changed from canon (the Lannisters) and others significantly left unmentioned (Jon Arryn, Hoster Tully, any of the rebel lords), somehow winding up in his current situation. Speaking only for myself, this train of events doesn't quite work.
*shrug*
I could be too insistent. It's your story, so feel free to write it. Just keep in mind things you might need to address going forward.