I don't think it's worth getting worked up over the naming of the music piece. It's a sinister theme that fits the mood at certain points of the game, and it just so happened to be named after one of the most significant scenes in which it is played. It's the same way for other pieces too - "Interrupted by Fireworks", "Underneath the Rotting Pizza" to name a couple. It's pretty irrelevant to the writing, really.

And Flying Arrow: There really isn't much in-depth history given for Shinra, just that it was a weapons manufacture that (presumably) developed mako technology for that use. Then when the war ended and there was no longer a demand for weapons, it moved to Mako Energy instead. It would be reasonable that the company had been a family company, since the President share's the company's name, and the Vice President is his son. We aren't told how many generations, though.