Yes it has brake lights and tail lights in one. It should turn on when you flip the light switch, and the brake light comes on when you press the brake pedal. Most models have a spring attached to the pedal that goes to a switch just above the pedal.
And as far as i know, they don't have a horn,stock unless one is added .
Do you mean while the engine is running, or just when you turn the key to the ON position? because the key has 3 positions. Off, On and Start (Engine).
Welcome to the board and Congrats on your purchase of a great machine. If you need any parts look here: https://www.kawasakipartshouse.com/oemparts/c/kawasaki_utility_2007_prior/parts