If Al Powell was on desk duty, why was he driving black-and-white?

Powell was actually on his way home from work when he heard the dispatch (I think he was at a gas station buying a Twinkie, actually), and since he was driving in that area he radioed that he'd check up on it.

This actually serves as an introduction to his character and shows that he is a dedicated officer who doesn't just clock out when his shift is over. It's sort of a character reference in that respect, which gives more weight to his dedication to police work and his desire to help McClaine.

It's a plot hole.

If Sgt. Powell had been involuntarily removed from street duty he would have been assigned to a role where he would not have a firearm on his person while on-duty or off, unless it was own personal weapon. And he certainly wouldn't have received a patrol car or been responding to call in one. Even the LAPD (Los Angeles Police Department) doesn't have enough vehicles to allow suspended officers or officers under some form of discipline to take one home in the evenings.

In order to explain Powell's presence on the scene of the crime, how someone could have contacted the police department, how Powell was allowed to remain on the scene and how he would be able to communicate with John McClane, the film's protagonist, he need to be driving a police car. Any other scenario would have made far less sense, as off-duty officers rarely are the first responder to a crime scene (especially a noise disturbance), they usually would be driving their own vehicles and those vehicles are rarely equipped with police radios.

It's a plot hole as:

1. A suspended or reassigned officer would not be allowed to remain on thescene of such an event.
2. Powell would have been ordered to immediately discontinue his contact with McClane as the LAPD really had no idea who he was. McClane could just as well been a terrorist used to misdirect the efforts of the police department.
3. Powell's cruiser would have been towed away as it had "evidence" (Marco's blood and DNA) on it from it having been used as a "landing pad" by McClane as well as target practice by the terrorists.
4. Powell certainly couldn't have used another officer's radio for the task as that would have required him to reset that radio to hear McClane and thus rendering it useless to receive or transmit necessary radio traffic.

So the inclusion of Powell's driving a black & white was a narrative device that creates multiple plot holes in the film.

The easy answer is some police departments give vehicles to all of their officers to drive so that when they are parked at the officer's home, there is a known police presence. It cuts down on crime. About the uniform/gun ... He's still a police officer and required to be in uniform no matter if he will draw a gun or not. He wears a gun because it is a part of the uniform.