I don't know what the (former / current / future) intended use of that outlet was when your buddy wired it years ago but it looks to me as though he had intended to separate the top and bottom halves into different circuits.
A similar kitchen application around here is the outlet under the sink. In that case you would have one hot for the garbage disposal and a second hot for the dishwasher, each served by a single neutral as you have in your box. In your situation your first clue may be the tag on the unused breaker (if there is one, a tag that is).
Now it is possible that one or the other of the 'hots' is common to other outlets in the area. In other words each half of that outlet doesn't necessarily have to have a circuit breaker that is unique to it. In my kitchen example, I believe that garbage disposals typically have their own circuit, while the dishwasher often shares a circuit with other kitchen appliances (or maybe the other way around??).
In such an installation, the electrician needs to remember to break out the jumper on the side of the outlet. Failure to do so results in just the situation you described where the breaker trips while the wire remains hot, even while disconnected.
The jumper is visible on the side of the outlet between the two screws. Grabbed with a pair of pliers, a couple wiggles back and forth and it will break free. For two circuits sharing a neutral, remove only the hot (positive) jumper, for two circuits supplied with unique neutrals (rare in my experience) remove both the brass (hot) jumper and nickle (neutral) jumper.
Once you identify the reason for the extra circuit, then you will know which half of the outlet you should pull your new line off of. Also, once you break out that jumper (if there is a reason to do so) then you will be able to properly connect the line at the breaker as was originally intended.
It is very possible that the extra circuit was intended for the microwave. In that case, remove it from the existing outlet, connect it to the line running out to the new outlet and then reconnect it at the breaker. Ideally in that case you should run a new neutral back to the panel but you could get away with tying it into the existing outlet. It wouldn't be my choice but it would work.
) is potentially hot. Couldn't hurt to hit the gray wire with a red permanent marker just the same. Better safe than sorry. That could be said for all the work in this box. It looks simple enough to me but if you have any concerns, give your buddy a call.