The main antagonist has been described as a cat by his designer, and while he does take on some damage throughout the story mode, he does not permanently die, and no longer resembles a cat (due to powering up/down) during sections where he is hurt.
No, but the end of the "Final Boss" level (note: name of song, NOT actual role in story!) implies this, with the opponent losing its grip on the walls, spinning out, and going off the bottom of the screen with an 'explosion' effect to a cheery riff. However, they're right as rain not long after, ready to inflict a different kind of trauma!
Nothing is explicitly said to be self-harm, but there's some imagery that implies it - one character stabs himself in the head with a triangle on two separate occasions to absorb the triangle's power. The same character accidentally injures himself earlier with his own attacks, resulting in his defeat.
At the beginning the main villain is shown destroying the credits, when Lachh's name is destroyed some text that says "I didn't make this D:" shows up.