This has been discussed a lot earlier in the thread. No, a scale and a key are not the same thing. One of the problems with the way we discuss modes is that we use the word mode for a couple of different things. A mode can be a scale but can also be a key (or "key," if you want to reserve the term key for standard major-minor keys). A scale is just a set of notes associated with a key. It's only half of what you need to establish a key, the other part being the tonal center or tonic (or final, in the older modal terminology). The same set of notes with a different tonal center is a different key. Again, this was discussed quite a lot earlier in this thread, so you might want to review that. No, G-G on the white keys is not a G mixolydian by definition, though that's one of the things it can be. And no, if you play a sequence of notes that looks like G mixolydian (mode = scale, just a set of notes), that doesn't automatically mean you're IN G mixolydian (mode = key.