I think the implication is there that Purslane might be Abigail, but it's never for sure as Abigail was very careful about all clones being equal. The reason she can't remember the name of the little boy is, I believe, because it relates to the Line's memory erasure of the First Machines. The flashbacks to Abigail's time in Palatial aren't for nothing, they're a parable about the First Machines. The First Machines were curious but they lacked creativity and they were fundamentally flawed, all possessing a core weakness the Lines exploited. This is analagous to the Ghost Soldiers in Palatial also having the same universal flaw that the Princess's magician exploits. Purslane's memory of Abigail's time in Palatial could be historically accurate, or her way of hiding the truth from deletion. She wouldn't be able to hide a name, but a fantasy story that parallels real events?
Whether the first machines arose from Palatial's programming or from the remnants of the little boy's mind after he was plugged back in for good I'm not sure. Given the fact that the little boy's name was deleted but not the name of "Palatial" is probably a good indicator for his mind becoming the First Machines. But either way, memories of Abigail's involvement with their origins would have been scrubbed to cover their crime/tragedy.
I do think that Valmik is separate from the little boy and Abigail's past, the author details his history pretty well, and it isn't contradicted elsewhere to show it might be fake. Valmik says he remembers all his past, even if those memories are buried, and if he truly was the little boy I think the author would have shown that connection explicitly.