We have considered different APIs for our weather widget and have come to the conclusion that Yahoo ! Weather would be one of the better choices. It is free for non-commercial use, accurate, worldwide, and not controlled by a government agency. The supported data formats are XML and RSS, with no JSON. These factors make Yahoo! Weather a good choice, as it was one of very few sources to meet our needs. However if we find a better Weather API we will consider it, this is in no way our final choice.
As for the music widget, we have not even begun to discuss the different choices for APIs. I know for a fact that Youtube has a Youtube Data API that allows interaction through .NET, Java, PHP, JSON, and some others. It's definitely worth looking at, I feel we could make a pretty cool add-on using this Youtube API. However there are a lot of others to consider.
Bugzilla is something that we will have to discuss more intimately with Mozilla. I'm not 100% on what functionality it should include.