I think that girl is Theo's sister

That's what I suspect too. While the shading of white is off, they both share the same eye color.
Either Violet's mom (who does have similar hair color to Lime) cheated on the runaway viscount or genetics are crazy.
But that could possibly explain why the mother was murdered, infidelity could have led the mother being killed.

Yeah this could be the reason behind her murder.

Exactly, also explains the erratic behavior the Viscount was going through with getting money from loan sharks and racking up debt for his business. He was going through impulsive behaviors probably after he or someone else killed the Viscountess maybe even before her death but it was hidden well until said event happened and every just went up in flames.