As ridiculous as it is, the tribble scene was actually the superior set-up scene because it, y'know, set something up which actually paid off later in the film.
The setup was so obvious, though, that it made the plot too predictable.
Well, the point in the story is Kirk makes his sacrifice unaware of what we (the audience) suspect could happen. If Kirk had been aware there was a chance he'd survive his trip to the warp core, that would've truly spoiled the story.
I don't think the point was to create suspense over whether or not Kirk would live, either, so much as show what Kirk was willing to do to save his ship and crew. Again, within the story, he's giving his life for the rest of the crew. That, and he can be afraid. That's the growth of the character.
It seems to me that the use of the man to bomb the Archive was not actually useful. Khan could have just used some kind of craft to fly above and drop something on it. Why did he risk the man just refusing to do it or telling his superiors instead? Seems illogical to me.
I've thought about this, too. As far as the father goes, as I stated above, he probably had to complete his end of the deal or Harrison would've killed his family.
Still, Harrison could've used a cyber attack, planted a time bomb, used a drone, or done any one of six or seven other things before needing a suicide bomber. Of course, the point of what he did within the story is that in the three or four minutes it takes the scene to play out, we learn just how cruel and manipulative Harrison is.
There's also the shock value of the bomb going off. That's probably the first time that happened in decades on Earth. Pure terror. Sloppy. If someone was willing to blow up what everyone thought was just a public archive, then what's next?