How to get this chest in the Abandoned Factory

Based on this guide, as well as my own experience, there is no way to skip the opening. In general, you can skip cutscenes by holding the Circle button, but this is not available at any point during the opening. From the article above:

Depending on how long you take, even getting to the first boss can take 30 minutes to an hour. Die any time before or during this fight, and you'll get the lovely screen below followed by a quick kick in the ass that sends you all the way back to the opening cinematic.

Dying repeatedly in Nier: Automata can tilt even the strongest-willed men, especially in this first segment, since you'll be driven all the way back to the opening lines, which, of course, bemoan the perpetual "spiral of life and death," in a cruel twist of irony (and the knife, for that matter).

As an aside, personally, to get past the introduction level, I had to bump the difficulty down to Normal. However, the guide above gives some useful tips on how to beat it without dropping to Normal difficulty.

Aside from analyzing the BGM and looking cool, the HUD Sound Waves chip serves one other purpose. According to posts like the one in this thread, it helps you find a specific Pod in the desert:

There's an item in the desert (pod C) that emits a high-pitched sound when you're close to it so it's easier to find. The sound display detects this noise, giving a hint to people who can't hear it (or hear poorly in general, or people like me who thought my speakers were dying instead of thinking it was an actual game sound).

This is backed up in this thread, as well:

I believe it's meant to help those who have a handicap with their hearing.

There is an thing (a pod) in the desert that can be found through a sound it makes. Not everyone can hear it.

If you've already found this Pod or are confident in your hearing abilities, there appears to be no other purpose to this chip other than being a BGM analyzer.