I think the moment when Claire asking for her father’s name would have made sense is when she phoned Meredith.

After that, it’s believable that Claire didn’t ask — purely because she was overwhelmed with sadness and confusion.

Meredith told Claire that her father didn’t care, and it shattered Claire’s hopes. I’m not sure learning her father’s name would have changed Claire’s reaction or made her any less angry. It’s not like Claire would have ever wanted to contact Nathan after she watched him drive away. She hated him for it, hence the broken windshield.
It’s debatable, though, and I totally agree that ‘Fate’ will bring the characters together at some point. Probably, as you say, through Peter.

I kept waiting for Meredith to give Claire at least the name of her father–yeah, she got the name “Nathan” from eavesdropping, but she never got the name “Petrelli,” which would have been of huge significance to her. Wouldn’t she have asked what her “biological” last name was? It seems to me to be a hole that she didn’t get the Petrelli name out of Meredith, but I think they must be reserving it for some big reveal involving Peter somewhere down the road. At any rate, I was somewhat disappointed that we didn’t get Claire’s reaction to that meaningful name. I guess it would have minimized the outrage with the rock-throwing and all, but still…

Yeah, good catch: at least now we know that Sylar was working from Chandra’s list.

Here’s a thought, though: Mohinder broke into Sylar’s apartment in “One Giant Leap”, and Sylar’s map had a ton of names which Chandra’s didn’t.

Not saying the GNs don’t correlate with the show, but that’s a plot point which implied that Sylar was independently tracking and discovering individuals. I think that part of the plot still needs to be resolved.