What happens to the companions after the battle of whiterun?
Nothing. They stay in the game to be joined after the quest.
By asking around in Whiterun. Usually you can just talk to a bartender in one of the inns and ask for rumors or information. This will give you the Miscellaneous Task "Join the Companions". You can talk to the leader of the Companions in Jorrvaskr, their base in Whiterun. Jorrvaskr is located in the small area in front of Dragonsreach. It is a large building to the east (in whiterun)
You can find Aela the Huntress in Jorrvaskr, Whiterun. She is usually outside in the back, talking. She is with the companions and if you join them, you will find her more easily. Or, you can find Aela just outside Whiterun, in a farm where she and two other NPC's ( Non Playable Characters) will be fighting a giant. Hope I helped
None of your companions can die if they are following you, but they can die if they aren't. An example would be if you are in combat and you tell her to part ways and she gets killed by the person or creature attacking you because she stopped following you. I used "she" because I am referring to Lydia, your bodyguard you get from Whiterun for being the Thane.
You can join groups or do the entire campaign, but joining groups is better because you can get other stuff like become a werewolf (if you join the companions in Whiterun, which I'm in) as well as earn a wage of about 100 - 500 coins a mission. (Note- If you do the campaign, a good way to get stuff is through picking things off of dead bodies)
The first battle with Jack, you just flourish him until he flies up. Then you shoot him and his companions with your bow. When he makes that light that attacks you, hide behind the rocks. The second battle with Jack, you turn on your life shield spell (if you're smart enough to get it) (I wasn't) and commence to wail on him and ignore his companions. It's that simple.
I think you are confusing two cities. Whiterun and Winterhold. 'Dragonsreach' is a keep accessable on the north side of Whiterun. The Winterhold mages college is also north of Winterhold, accessable via the long bridge on the left as you enter the town. You must pass a magic test with Faralda first though.