1. Story. Is it different for each class? Is this story/Are those stories any good? I mean, I understand that it can't have the depth of KOTOR II, but still.
Yes the story is different, though the quality of stories can vary greatly from class to class. Most people say (that I've heard anyway) say that the Agent storyline is the absolute best, as it apparently has the most nuance to the plot, and more depth to it. Many people feel that the other storylines were done after Agent, and suffered from reduced resources and deadlines, but I have no way to verify this.
2. Companions. How good/bad are they as characters and gameplay wise?
They are very good now with the newest expansion changes. You can alter their combat role to suit your own needs (Tank/Dps/Healer), instead of being forced to use a specific companion to fit a specific function. Plus, now that their stats are no longer governed by the gear they have (all gear is now cosmetic for them), they are pretty freaking OP in my opinion.
3. Will I need to grind?
Newp, not at all. They've drastically increased the xp rewards for class storyline missions, and for the planetary story missions, (the mission icons that glow purple on your map, those are the really juicy missions). The other random missions are still there, but people who have already experienced that content with a previous character of that faction, usually skip it for the more time efficient purple missions.
My opinion on content up through Hutt Cartel expansion
I personally find the content to be somewhat bland a lot of times, mostly because the writers had to have the final result of the various missions end up the same, whether you go Dark Side or Light Side. So, for example, when given a mission to save this planet from an evil Sith warrior, whether you A) Do the good thing, and defeat him, rescuing all hostages/slaves at the cost of valuable resources to the Republic, or B) Turn the Sith warrior to your service as a minion, sell his slaves for profit, and hire mercenaries to pilot a fleet of starships as your own personal armada, the end result at the end of the day, when you stand before the Republic representatives is that "You were a fine and noble champion of the Republic, who is worthy of praise and honors!" It makes the various factions seem completely oblivious to what goes on around them, which I find personally annoying. Your individual actions have ZERO direct impact on the actual storyline itself.
Now, the above statement does not refer to the newest Knights of the Fallen Empire content, which they have hyped as "having choices that actually matter!" I personally don't know how accurate this is, as I have not reached that new content yet with my characters, but if it is true, then I would find that content very interesting and enjoyable indeed. Regardless, I do find the game fun, especially now that I can play it solo with my companion, and not miss out on that much.