Can I ride the unicorn? If not, can I kill the unicorn?

Definitely invulnerable and nothing more.

Shooting a unicorn:

this.game.physics.arcade.collide(this.boss, this.ammo, function(t, e) {
    e.kill()
}

does the same as shooting a platform:

this.game.physics.arcade.collide(this.platforms, this.ammo, function(t, e) {
    e.kill()
}

i.e. destroys the projectile but not the object shot.

Furthermore, any collisions with the unicorn simply end in death:

this.game.physics.arcade.collide(this.player, this.boss, this.loseGame, null, this)

Running into the unicorn will kill you and shooting it pushes it back slightly. As far as i can tell you cannot kill it. I have currently shot it over 600 times in a row and it is still walking towards me. Apparently unicorns are invulnerable.

The box art and menu art assets were created in parallel with game development, but the game development changed course without coordinating with art, so there's a slight discrepancy, and you cannot kill or ride the unicorn.