1. I feel that "Media" is not the optimal label. I understand that it's supposed to include images, audio, and video, but I suspect that the way it is now, it's equally unclear to many people for all the types of media. For years, every time I was showing people in real life how to edit a page in VE, they asked how to add an image, and I had to tell them how to find the "Media" button. Many people had trouble finding it themselves. Perhaps something like "Images and media" would be better?
I believe that originally it said "Image", and people complained about it being wrong, so we changed it to "Media", and haven't had any subsequent complaints.
Was any focused user testing done about this?
Not about variations of the label itself; we asked brand-new users in our regular user testing to insert an image and they succeeded, but that doesn't mean that they wouldn't do better/be happier were the label to be changed.
That said, "Image and other media" would be very clunky and lengthy, which undermines a big point of there being a welcoming, intuitive toolbar. :-)
Back in 2012 there was talk of making "media" a first-level item in the toolbar (i.e., directly in the toolbar for most users, not in the Insert menu) to encourage users to make richer, more pleasing experiences for readers. That said, I don't think that's currently planned.
2. Should the 2010 WikiEditor label be the same as Visual Editor? Functionally they are quite similar, and "Embedded file" is definitely not optimal. Both "Embedded" and "file" say nothing about images, and are quite technical.
We've been very wary of changing the 2010 wikitext editor experience to avoid disrupting existing editors who are used to the current set-up. I'm not sure this would be the place to start changing that stance, but if it's a compelling change we could vary that.
3. Is it worth doing anything about the 2006 labels, or are they just going to be completely removed from the code soon?
I don't think it's worth it, no.