1. Yeah, the PvP mission terminal gives you missions with warzone rewards and the like. The PvP isn't actually on the Fleet, though, you just accept the missions there. To enter a PvP match, you need to enter the PvP queue, the button for that's on the map GUI (if I'm not mistaken, it's the Empire/Republic sign). A lower level gets teamed up with other low level players, so you don't find yourself fighting 50s (as it was at the start).
2. There are customization vendors on certain planets. I know Taris and Belsavis had one, there might be more.
3. /gquit in the chatbox. A shame you're not on Harbinger...
4. Most if not all emotes can be found by clicking on the little figure in the chat GUI. There's quite a lot of them. You can also create your own emotes with /e "text" (without the "").