Pokemon-go – Why is the gym icon color on the Pokémon different than the current team

On average, there is a slight disadvantage to picking the weaker team, but not one that will completely affect your game.

For most of the game, your team affiliation will either not affect you (spawn/catch rates). There used to be rumors about giving you a perk for that team, but there's been no real evidence since. (Even after the game was reversed engineer to see more of the "gears", like how many steps to make a spawn call.)

The disadvantage comes from the lack of opportunity to earn the Defender's Bonus that Gym Leaders get for having a Pokemon at Gym. If you join a team that is the weakest in your region, then you and your team will have issues in holding onto Gyms. (whereas the teams with more members can statistically attack, take over, and fortify easier.)

The biggest takeaway though is that this is only an opportunity cost. The weaker team might be stronger a mile away, the stronger team might not frequent the area you are as much. The number of variables are unlimited as to that the weakest team might control all the Gyms of a city since their members are organized. There's no hard-set disadvantage, just a fluid difficulty curve.

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.