I've always assumed that Nan Balat was unconscious during the time she used/picked up the shardblade.
Yes the shardblade could be used after death to cut the fathers body (and thus the soulcaster), but it seems unlikely that Shallan or anyone else would have done so. It seems equally unlikely that it was done by accident especially if the shardblade belonged to Shallan's father and so would appear next to him as he died. I suppose he could have already summoned it but not had time to use it. However as the corpse in the picture is described as lying face down it again seems unlikely that the blade could have cut the soulcaster in his pocket as he fell. Not impossible though.
We know that Shallan normally has to blink and deliberately take a "memory" of a scene which she then translates to a charcoal picture. We also know that in times of stress/trauma (e.g. the attack in the alleyway) that she remembers these scenes vividly without having to do the blink technique thingy. I could see Shallan doing something similar as she killed her father and immediately after she killed him, but I don't see why she would deliberately memorise the scene later especially as she seems to want to forget about it and repress the event as much as possible.
I could be wrong about the furnishings of House Davar. However nothing I've read so far leads me too the conclusion that you would describe them as lavish. House Davar is a minor house and Shallan describes the estates as being isolated and backcountry. Now I'm not saying that house Davar was especially poor or that the house was threadbare, just that the impression I get isn't one of lavishness. For example Shallan states that the library only had 87 books. Of course I could be wrong and that her father spent a lot of the quarry money on doing up the place and not on books etc.
Shallan having a shardblade prior to her Fathers death is of course the big, gaping hole in the theory that a shardblade killed him. Where on earth could she have picked up one and why would she, as a woman on Roshar, pick it up? The initial logical assumption, especially as she calls it the fruits of her sin (or something like that), is that she got it from her Father. The problem with that theory is where did he get it from? It seems unlikely to be a family heirloom (shardbearers write their own ticket most places on Roshar so house Davar would probably be quite influential and not the other way round). It also doesn't really make sense for the Ghostbloods to give him one as well as the soulcaster, especially as if her Father had it, it was kept hidden and not used. This to me would make no sense, unless they have so many shards and plate that they can afford to give one as a bribe to hold Shallan's father to them.
There are other possible origins for Shallans shardblade though, although these are pure conjecture and backed up be no evidence whatsoever. The first that comes to mind is that the blade was in the hands of an assassin sent to kill Lord Davar and that Shallan somehow ended up with it. The other possibility that I can think of is that the theory that Knights Radiants can create their own Blade and Plate is correct and that Shallan was able to do this somehow. However, while I like that theory because it would be so cool, it makes no sense for Shallan to be able to do so as she has only just started down that road since she reached Karbranth.
In short I just don't think we have enough information to say where Shallan's shardblade came from, except that she feels she got it as a result of her Fathers death at her hands. Anything else is pure speculation at the moment.
Sorry for the essay. Hmmm... I wonder if I should add a bibliography.......