Hey all, this is my first post here so if I'm out of line, please be gentle.
I'm not too crazy about any of the current theories on Shallan's father's death and how she acquired a Shardblade so I'd like to present my own theory.
Based on Shallan's reaction to Jasnah's murder of the theives, and her complete aversion to any type of confrontation, it doesnt seem plausible to me that she has ever killed in the past, even by accident. ESPECIALLY if she hed killed her father. She seems too soft spoken, timid, distraught and emotional. She didn't comprehend how someone could take another's life even when threatend by four knife-weilding brutes in a dark alley. And all this even though she completely understands that "The men Jasnah had killed were terrible creatures, and she had little doubt that they would have killed her." Plus, taking into account that she completely froze during the attack, remember she is described as being paralyzed with fear, how could she have attacked and killed her father?
My current theory is similar to some others here, in that I believe there is a third player in the death of her father. We are all missing (or overlooking) some important aspects of what happened, but it's the best solution I can come up with. Here's my thought on how it might have happened;
Ghostblood visitor is conversing secretly with father and possibly Nan Balat, perhaps trying to recover the lost Blade and Plate given to Helaran (which is another theory I believe is correct). Meeting goes poorly and visitor decideds that Nan Balat might have the items (or perhaps just to coerce more info from Father) so he begins torturing Nan Balat for information or compliance. This would explain the shattered leg and how near death/unconscious Nan Balat was. Somehow able to get close to the visitor, Father uses the fabrial to change the visitor to smoke, but not before being fataly wounded. Blade materializes, slicing fabrial, perhaps also killing father.
At some point during this event Shallan, hearing the commotion, enters. Perhaps this was when Father had his chance to kill visitor, perhaps after Father had already vanquished him and been wounded. Father orders her to give him the Blade, or give it to Nan Balat, or even perhaps for her to keep it. She picks it up, the knowledge of it's use infuses her causing her to drop it, and it turns to mist. Father slowly bleeds out his life while Shallan watches. The important part is that Shallan sees her father dying and instead of helping him she stands by and lets him die. Maybe the knowledge infusing her put her into shock, or perhaps she saw it as a way to end the family's torture. Whatever the reason, she let him die and that's why the Blade is the fruit of her sins and something she never wants to use.
Right now, based on Shallan's current character, this is the only explanation that makes sense to me.
