Exterior

Hard to be sure, but it may not be the pipe. That's the really frustrating thing with water, it travels. Is that a skylight beside the pipe? The initial leak could be there and the water migrates over to the pipe and slides down that. It could be that you have a hairline crack that you really have to look to see, it doesn't take much. Really look at the flashing around the pipe and seal anything that remotely looks like it might be an opening. Unfortunately, sometimes with leaks you have to start with what seems the most logical source of water and fix one thing at a time until you get lucky.

Go to store. Buy masonry sealer. Go home. Paint on brick. Let dry.

this should be handled through you board president and the management company. let them negotiate this to get it for free from the developer. if that's a pain, then get a contractor and do an proper analysis and fix it yourself.

I'd go with the copper gutter, the half round style is standard and will be the cheaper option. But, for a nicer look, but more expensive, use a copper gutter formed into the "K-style" or ogee type. Even a non copper brown aluminum gutter of this shape will look better. http://slateandcopper.com/gutter.html?gclid=CIz7l6LC9qgCFU5qKgodkU6jUg