Pokemon-go – Cannot fight at Gym

Most (if not all) of the problems listed in the question are related to the Pokemon GO servers being offline, slow to respond, or overloaded with requests.

In the meantime, nobody likes not playing the game. Fortunately, there are a few things you can do:

• Just Wait
  The easiest (and best) move to take is just to wait for the servers to restore to a normal state. You can check the server status at this handy page, or by just trying to log in to the game.
• Force Quit the App
  Sometimes, the app will get stuck in a cycle of failure and will be unable to actually do anything. You can fix this by [iOS] opening the Multitasking window and swiping up on Pokemon GO, or [Android] navigating to Settings > Apps > Pokemon GO and then touching the Force Quit button.
• [Android] Clear Caches/Data
  As a more extreme measure, you can go to Settings > Apps > Pokemon GO and tap the Clear Data button. Note that this will only delete your local data -- your Pokemon will be safe. However, you will need to log in again. Be sure you know what account you used/created for Pokemon GO if you do this.
• Reinstall the App
  If all else fails, you can just go the nuclear option and erase the entire app and start again. If you have an existing account (and you know the credentials used), your Pokemon will be at no risk and will all be there once you log in again.

Can multiple players of the same team fight their own team gym in order to raise its prestige?

Yes, however it is not like when multiple players team up to take down a gym. The training can be done simultaneously but in separate battles. Each person's training has no effect on anyone else. As if the training was being done one after the other instead of at the same time.

If you have any experience with this, then can you also include the amount of Prestige you gained for your gym and the CP of your attacking Pokemon versus the defending Pokemon in the gym.

As is outline in this answer; you get 500 prestige for defeating a defending Pokemon with equal CP values, 100 prestige for defeating it with a double or more Pokemon and 1000 prestige for defeating it with a Pokemon half the CP or less of the defending Pokemon. For CP values between half or double it will scale somewhere between those values.

What would you recommend, fighting your own team gym alone to raise it's Prestige or fight with a group (if it's possible)?

It's exactly the same either way for gyms of I think level 5 or less, but above that it becomes better to do it alone. The reason for this is that with more people helping you raise the prestige, the more levels you need to increase the gym by in order to place everyone's Pokemon in the gym. For example with two people you gain prestige twice as fast but need to level it up twice us much. If a gym was level 2 and needs 2k more prestige to the next level, you will need another 2k (after adding a Pokemon) in order to place a second Pokemon in the gym. But at higher levels this will change to be considerably more prestige so it is easier to only raise it by one level instead of multiple.

That being said, higher level gyms are harder for the enemy to take down and thus can be worth the additional effort.