Yes. In the second pilot, the first one with Kirk as the Captain, they left the galaxy. In the one with the Medusan Ambassador, they left it again. I am not sure if the Kelvans took he Enterprise outside the galaxy or not (the episode in which the crew was mostly turned into small paperweight-sized objects). And in one episode of TNG the Enterprise D wound up so far from this galaxy, I don't know if they could even tell where it was. I think this was a Q episode but am not sure.
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In Star Trek series and movies, there is no canon evidence of Starfleet vessels traveling beyond the Milky Way galaxy. The technology and distances involved in intergalactic travel make it very challenging. Most of the exploration and conflicts take place within the Milky Way galaxy.
I can't determine what you're asking about. Stars do travel; they orbit around the center of the Milky Way Galaxy.
I said WHERE?!
The sun is but one of several hundred billion stars within the Milky Way Galaxy. Thus the galaxy is much larger. The Milky Way is so vast that it takes light approximately 100,000 years to travel its diameter, and 1,000 years to travel its thickness. http://en.wikipedia.org/wiki/Milky_Way
Star clusters are typically located within the Milky Way galaxy. While there are some star clusters that are closer to Earth than other parts of the Milky Way, overall the Milky Way galaxy contains a vast number of star clusters that are more spread out.
Planet Earth belongs to the Milky Way galaxy, which is a spiral galaxy containing billions of stars, including our sun. The Milky Way is vast, spanning about 100,000 light-years in diameter. Earth is located in one of the galaxy's spiral arms, known as the Orion Arm or Local Spur.