On Shallan killing her father... I certainly have plenty of questions.
Shallan hates conflict / people shouting etc. There's no indication that this is recent. So if her father was beating up one of her brothers in a rage, she would likely flee if possible or probably sit in a corner hugging her knees or something. Was she sufficiently provoked that she overcame that? Possible, but seems a bit convenient - though possibly after taking action once, she became even more fearful of conflict.
Without a good weapon, how could Shallan kill her father? She should be far weaker than him. She has no weapons training or training in combat either. Her father should have both. So for Shallan to have been able to kill her father in a violent rage does seem unlikely.
Did she actually kill him or just feel responsible? Her brothers don't seem to treat her like a killer in the bits we see.
There's also no positive evidence that I remember that her brothers know she has a Shardblade. If she jumped in to save one of her brothers and killed her father and got the Shardeblade there, then they should know. As for not selling the blade maybe they're so ludicrously expensive that they basically can't be sold without huge risks - maybe selling it would be a last ditch option.
On a side note, could Shallan even wield a Shardblade? I seem to remember them being quite heavy - works well with Shardplate because of that. Can't find reference to weight right now though. Szeth's one is unusually small and he also had Stormlight to boost him. That being said, Shallan seemed to think she could use it, which is why she tried to summon it at one point - that she shouldn't bother trying to use the Soulcaster since she doesn't know how to use it but does start summoning the Shardblade instead, even though she promised herself that she wouldn't use it. So maybe she has used it before.
Enough hints to speculate, but far from enough to reach any firm conclusions.