Does it need electrical installation or batteries?

This doeasn't need any electrical or batteries.  The faucet comes with smart sensor.  The LED lights will be generated through the flow of the water, and the color of the light will depends on the temreture of the water.

Water sits in the tray after turning water off. Is this typical or did our plumber install it wrong?

There should be no water residue in the tray. The tray is about 1 degree inclined

How do you turn this on and adjust the temperature? Where is the handle/lever?

This is turned on/off from the square knob on the top of the faucet (behind the ball)