A lot of the time, especially on certain planets, when I use the Galaxy map (not from the ship) to travel to a planet, after the loading screen instead of appearing on the planet the game exists to the character selection screen. When I login back to my character I am where I was before I tried traveling to the planet. If I go to my ship and use my ship to travel to the planet it avoids this problem. This happens a lot when I travel to oddessen and dromund kaas. Has this happened to anyone else?

You can play every single mission with each other regardless of class (excluding starting planet missions of you don't start at the same planet, but that's not relevant to your case). If you are a different class, you can still participate in the other classes conversations during class specific missions, but you won't be able to actually add to the conversation (only whoever's quest this is can talk during class story missions). Since you're both playing Jedi warriors, you have the same class story, but I'm not sure whether or not you have to do the conversations twice or if you can both talk during conversations. But you should be able to quest together regardless.

I usually farm flashpoints (Hammer station is a good one) or do heroics. If you just find yourself doing the story and not having enough xp to continue to other planets, I would suggest playing a couple of GSF matches now and then between missions. Its fun and you get a lot of xp from it.