Heya, this is useful feedback, thank you!
In terms of the story, what would you like to see for it to be more interesting and fresh?
In regards to the music, this was all composed for the game with the intent of having a similar musical style. Does the rest of the music feel equally out of place, or is it only those two tracks you mentioned?
As for your thoughts about combat and UI, here are thorough explanations of each of those elements.
Speed/Wait are percentages of the gauge. If the skill Speed is 50, then after using the skill your action gauge automatically fills up 50%. If the Wait is 50, then after selecting the skill the character will wait an additional 50% of their gauge before the skill executes.
You are correct about skills that cost Infinity TP; they always have the same power and cost your entire gauge, so ideally you want to use them when your TP is as low as possible.
Infinity Beat means that this skill has no Finisher and (in theory) builds up Beat infinitely. Of course, since the only skills with that feature in this build are non-damaging, they don't build Beat at all. In contrast, a skill with 0 Beat always meets the criteria for its Finisher, which means that the skill is effectively always a Finisher no matter how much Beat you have.
The darker, faded portion of an enemy HP bar is displaying how much HP they had before the most recent attack used against them (in other words, illustrating the relative amount of damage that attack dealt).
These aspects aren't all explained right away, in part because we trust the player to pay attention and make educated guesses about certain mechanics (you were pretty spot on with your inferences!) but also because many of these mechanics will be explained more thoroughly when the time comes. Most of Matsumoto's abilities are intended as a sort of "preview" of what's to come, as the game slowly introduces these abilities to the other party members in a slow and natural manner.
With that said, it's not at all the intent to confuse players, so this feedback leaves a difficult balance for tutorialization between overloading the player with too much information, and not providing crucial info they'll need to enjoy the game. I'll have another look at these elements and consider if there are other ways to hint at the mechanics during this prologue section.
Thanks a lot for playing, this was very useful data!