Does the kitchen sink need a P-trap with a disposal installed?

Yes. It prevents sewer gases from entering by way of the water trapped at the elbow. Gases need an upflow which is eliminated by the P-Trap. It's not a pee trap, it's a p-trap. Yes you do need a p-trap. The p-trap holds some water in the trap to hold back the sewage gas from coming up through the sink. IN ADDITION: A P-trap can be used to unblock or unclog small blockages either in the drain pipe exiting the trap or the trap itself, because it holds water, by covering the plug, it works the same way as a plunger does and by squeezing it continuously it forms a suction drawing or pushing blockages, freeing them and pushing them further down the line until it is clear. It is useful to know when a plunger is not at hand.