Obviously, this is going to be full of spoilers for the endgame. I'll try to break them down so that you can minimize your exposure, should you wish. I'm not going to go into heavy details about story (except where noted), just how the gameplay mechanics shift.
After you use the chamber, the effects at the base are:
You can no longer "scan for activity" (ie, advance time) on the mission control view. Therefore, any research or foundry projects will halt, and soldiers will stop healing. You can't get any further money or resources, although you can build items if they are completed immediately. The only mission you can take is the final assault.
The soldier who uses the chamber:
Will gain a new Psi power - Rift, which has a four turn cooldown. It generates a damaging Psi field in an area for 2 turns, which will severely damage any enemies inside of it. It has a range similar to the rocket launcher, so if you plan on using it on the final mission, you should probably put it on someone who isn't going to hang way back like a sniper. A front line trooper will probably get the most benefit.
When you accept the mission:
You must take the soldier who used the chamber. If this soldier dies on the mission, the game is over. The mission itself is fairly long, so pace yourself. At certain points aliens will spawn from nowhere, so be on your guard as you move up - you will be flanked as if by magic. Expect to fight at least one of every alien, and at times multiple of the more challenging ones. The ship is broken down into rooms, and you must kill all the aliens in a room in order to proceed.
After the mission (massive story-related spoilers!):
There is no chance to continue the game further. Also, the soldier who used the chamber will die, for what that's worth. Since the only thing left is to watch the ending and the credits, this is not a big deal game-wise, but if you have a particular attachment to your squad, pick someone knowing that they're not coming back.
If you capture several Outsiders, you will have all the shards in your inventory. But the most frustrating thing is that you will not be able to sell the shards. So you will have the shards stuck at your inventory for the rest of the game.
Outsiders will stop appearing only after you finish the mission that comes after you research the shard, but you still won't be able to sell the shards.
Since they are harder to capture that other low level aliens, but have the same weapon (light plasma rifle), I would advise capturing floaters or thin men instead of outsiders if you want the weapons, as they will not appear in your inventory and reduce clutter.
Best Answer
Just like in the original, one Alien Containment is all you need.
If you capture two aliens on one map, both will be available for interrogation.