There is a problem with that though, and it's an inconsistency in the series. Remember back in Blood Rites when Eb said he tried to kill Lord Raith and couldn't? This was Blackstaff talking right? Margaret was able to block Raith's feeding with her death curse, but she couldn't kill him with it. Harry sensed the cold emptiness inside Lord Raith, he wanted very badly to kill him, but didn't try it because of what Eb had told him.. Fast forward to 12 Months and Eb tells Harry in effect that he chose not to kill Lord Raith? The point I am making is if the general knowlege was that Lord Raith couldn't be killed, that wouldn't satisfy Etri. Plus, even at this point, though most know better, Lara is still the power behind the throne not the head of the White Court, and if Lord Raith was handed over, that would blow the court up politically and destabilize things more.
He did try, but his options were limited, his political power is limited, or was. He did come up with a solution, getting that boon from Mab.
I doubt that, she had negotiated it with Etri ahead of time. Because if it was last minute as you say, there was no gararantee that Etri would have accepted those terms. She didn't set the price, Etri did.
I am not sure of what you are trying to say here. 1] Would the child even have been born? Did the Outsider want it to be born? Remember the mid-wife in this case was actually Mother Summer, was that necessary? If it was a normal birth, there were several people on sight that could have assisted. The fact that Justine was still possessed by the Outsider meant that it took someone like Mother Summer to assist and keep the child safe during the birthing process.2] Yes, if there was no negotiated settlement, Thomas surely would have been killed. 3] Why would Harry and Lara fight over the baby? They are still on track to marry if Mab gets her way, and she usually does, they would just be foster parents..
Personally I don't think the baby could have been born any other way. You don't ask Mother Summer to be a mid wife if it's possible any other way. Either the Nemesis would have possessed the baby to use as an escape route or as a vector for possessing others.
2) the solution to Etri's demands
Finally hours of obsessing over a song of ice and fire, and history finally pay of.
Mab properly didn't discuss it with Etri beforehand but Etri conditions were clear a life for a life. Either Thomas or someone of his blood. The baby is of his blood. Giving your child as hostage or a price to your foes to ensure peace or as a weregild was not uncommon in past across multiple cultures. Hell marrying Margie to Etri's son was not out of the question. Mab implies it although Harry sees it as a threat against Margie's life but I think this was what she was trying to get across. In the most Mab way possible.
Also Etri didn't have a choice in this situation. Someone had to be punished for the attempt on his life. Doing nothing makes him appear weak to both his allies and enemies. Even knowing about nemesis does not excuse Thomas. Thomas was not possessed he was forced to do it sure but he still did it of his own will even if under duress. He had to be punished and publicly at that. Etri taking his son is that. Then to show what a kind, noble , honourable and wise ruler he is to his people and other the baby has to be treated well to show that he is not driven by revenge and he can be a good Ally. Think Theon in asoiaf