One of the main differences in the Grand Finale mission that depends on which team you chose are dialog changes with the homies. Each team has different dialog for each of the mission parts. There are no significant gameplay differences that I noticed.
In the second part of the mission, in the part where (quoting Saints Row Wiki) "things from those homies nightmares come through but are friendly" - this will depend on the team you chose. It's not a big effect on the gameplay though, as you will leave that area shortly afterwards for your next objective.
Also, a mission part might be easier depending on which homies' loyalty missions you have finished (which homies have been transformed into their super-powered versions in the simulation). Super-powered homies will make finishing a mission part a little bit easier.
If you haven't finished all loyalty missions before starting the Grand Finale mission, IMO, you should prioritize choosing the super-powered homies that you currently have for the second part, and then the first part. In the third part, your homies will not be using their super powers, even if they have them. While the first part does not really require much from your homies (they will mostly just be driving a vehicle).
You can only pick one team per task / mission part. Once chosen for a mission part, a team can not be chosen for another mission part. I recommend saving before starting the mission, if you want to replay the mission parts with a different team.
Game Ending
Another factor that could affect the outcome of this mission is which homies' loyalty missions you have finished before playing the mission (rather than which team you chose). Because about this mission's ending...
There are two endings to this mission, though it is not through anything the player does in the mission. If all loyalty quests are completed, then the player will unlock the true ending where the Saints discover time travel and the narrator will be revealed to be Jane Austen. If not all the loyalty quests are completed, then the player gets the non-canonical "bad ending" of the game in which the Saints do not discover time travel and the player will not get the "Saints and Sensibility" achievement.
Source: Saints Row Wiki, "Grand Finale - Trivia"
Essentially all the non-loyalty side missions from all the characters will consist of nothing but completing the activities and targets already marked on the map. Kinzie's will always entirely consist of Store Hacking, whereas the other characters' side missions will be more varied. Ultimately, completing every side mission will require you to complete basically everything on the map, though you will only need to get bronze on the activities.
As to your second question, no, you cannot get the sidequest's rewards just for doing the activities/targets on your own without the mission. You can however do the activities/targets long before taking on the mission, and complete the mission in basically no time at all once you take it. At most, you'll need to head into the simulation, walk around a bit while people talk, then return to the ship to get the reward. You can even get partial credit; if a side mission has five activities in it, and you've done three before taking on the quest, then three objectives will already be complete, and you'll only need to do the other two.
Best Answer
It doesn't matter.
If the experience is the same each time, the first two explode and the third one works fine.
You pick one; it blows up and you need to pick from the remaining four. You pick a second; it blows up and you need to pick from the remaining three. (All the while being ridiculed from Kinzie.) The third pick will work out fine.