I agree with you except the part where Law didn't kill his parents. I am not sure, there is a fifty/fifty possibility.
Wow I didn't even consider that it could be Jesse's mom who kills Law's parents! I think that makes more sense to me right now. Jesse's mother is definitely very calculating and manipulative. It's pretty clear she would do whatever she felt like to benefit her family, i.e. her bribing Law to stay behind Jesse's back. I still see Law as too much of a cinnamon roll to kill his parents right now. Maybe she'll kidnap Law and kill his parents making him finally go over the edge and we'll get the fruit loops Law seen inside moritat. (But this could all be totally wrong ヾ(❀╹◡╹)ﾉ~ I should probably give up trying to figure it out)
I kept waiting for them to kiss after that rainy hug. That would´ve been the most romantic scene out of all yaois combined, especially considering Law´s and Jesse´s love-hate relationship. A kiss like that would be very meaningful and memorable since we all know that everything is gonna go downhill from here. I can bet any amount of money that this author is a sadist.
Jesse´s mom has given Law consent to stay by her son´s side and is willing to give compensation for his loyalty and friendship so Law does not need to rely on his parents anymore for warm food and a roof over his head.
Jesse is the only person to have ever celebrated his birthday and is the only one who actually wants him. Now that his mother betrayed him completely, the place he wants and needs to go to is the Brown family. Law realized that is the only place he belongs and where he is wanted/appreciated without being treated like a slave. I highly doubt Law killed his own parents. No matter how ill-treated he was, he never gave any indication of wanting to harm them. My guess is that Jesse´s mom killed them.