I believe each class has one. From what I've seen it is usually your first companion that you have quests to do with them. Though honestly I wouldn't mind doing companion quests with the rest of them.
i think it depends, cause with the JK, it's Kira, not T7, that gives you the quests.
i would highly suggest unlocking the legacy speeder skills on your alts. being able to use a speeder at level 10 makes the first few planets sooo much easier. i would also unlock at least one level of the field repair droid as that comes in really handy at times. i second the rocket boots unlock. really helpful.