1) What exactly was the point of the Noel Clarke scene? I understand it was to save his daughter, yet he then blows himself and the Data Archive up like 5 minutes later. Was it to show that the regeneration blood worked, which we saw with the Tribble later in the film? I was a little confused on why he was in the movie.
Yes to the regeneration question, but he also was the one who transmitted to Admiral Marcus about who made him do it.
It was almost the movie in microcosm. Is there anything you wouldn't do for your family? As far as why he did it goes, I'm sure if he hadn't followed through on his end of the deal Harrison said he'd kill his family (including the dog, this is Khan, after all). He was desperate to save his daughter, basically made a deal with the devil, and the devil took his due.
The scene also prevents anyone from saying at the end of the movie that Khan's "magic" blood came out of nowhere as a plot device. Its healing properties were established in that opening scene. So, the last part of the movie is saved from being a cheat.