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.
I would suggest that a lot of what you think you know, you don't know.
For instance, as a single counter-example, if she killed her father to protect one of her brothers, but this was because her brother had been beaten senseless and was unconscious at the time, then she could have gotten the Shardblade without anybody knowing.
Also, if her father wasn't
expecting her to attack (if that's what she did), she would have a huge element of surprise, especially if she did it from behind. Training becomes moot in circumstances like that, especially if it was a lucky blow.
I also would not be at all surprised if Shallan's shock at her own murder (if that's what it really was) actually drove much of her hatred of death and murder later. People are weird that way, and it makes sense to me emotionally.
Thus we don't know what happens, but what speculation we do do should be based only on the bits of evidence that have actually been given, not vast extrapolations that assume things we really cannot know.
These are the things we know for certain and that seem relevant:
(1) Shallan thinks of the blade as the
fruit of her most heinous sin. Simple grammar and definitions of the terms suggest strongly that "most heinous sin" leads to (in Shallan's mind) "Shallan gets shardblade".
(2) From what she tells the "truthspren?", she honestly thinks she killed her father, a "deep truth" that deepens her bond with the spren.
(3) From her discussions of morality (and reactions to the, um, "philosophy lesson"), she thinks that baiting and killing random thugs planning on killing and raping passersby is wrong; from this we can conclude that she abhors murder, both intellectually and emotionally.
(4) From (2) and (3), Occam's razor strongly suggests that "killing her father" is by far the most likely referent for "most heinous sin," especially since she's never had much chance to do anything else she considers wrong. Emotionally, I find this conclusion almost unavoidable.
(5) Thus we get "killing her father" leads to (in Shallan's mind) "Shallan gets shardblade".
Most people agree with this logic, but then argue that she may have gotten the Shardblade from something besides her father on that night. Possibly true, but extraordinarily unlikely. My logic with this is:
(1) Her brothers apparently know nothing about what happened, either the murder (if that's what we would really call it; it might not fit our legal and moral definitions) or the shardblade. This strongly suggests they were all indisposed or absent during what must have been a crazy night (that, no matter what actually happened). Thus what her brothers think is irrelevant except as negative witnesses. The most important things we know are things they didn't see, which constrains what happened, but only very slightly. My key point here is: Something crazy and deeply unlikely happened. The brothers seem to know nothing about the oddest parts of it, and don't suspect Shallan of being involved. Thus it apparently happened in a way that seemed mundane to them. Thus is probably happened fairly quickly, or at a time when nothing was expected to happen.
(2) We know that Shardblades are perfectly hidden from normal people until they are summoned or their owner dies. Taking a shardblade is as simple as grabbing it, given the chance.
(3) Thus father dies with Shardblade -> Shallan gets Shardblade by grabbing it out of sheer curiosity/complete shock-> it disappears, she claims it, and she wonders the on Earth she's going to do with herself -> it happened so quickly her brothers never notice -> she promptly runs away the best way she can, making reasonable excuses (perhaps destroying the Soulcaster herself? It's a good reason to leave, after all),--- sounds like an extremely plausible sequence of events.
There are a lot of details not covered by the above theory, but since whatever happened must have been strange, this is the least strange version I can see. Most importantly, said details can be shoe-horned in without much effort and without bending the basic outline. Most efforts to deviate from this simple outline almost instantly run into difficulties or unneeded complications. For insrance,
Anyway, that's how I see it.