That little bag holds your expended air and new air in reserve
Although glockshuter was a troll and was banned, it's worth pointing out that he or she was describing a rebreather mask, which isn't found in an aircraft.
The bag on an aircraft mask is a prebreather resorvoir; it doesn't take in any expended air, but only serves as a resorvoir for oxygen flow coming from the aircraft system. It's a non-rebreather mask; it doesn't take any exhaled air from the user. Only oxygen flows in through that bag. If one were to plug the mask inlet, or have a tight seal and a good valve on the mask, the bag would inflate. Non-inflation simply means the user is consuming the flow before it has a chance to inflate the bag. Not a big deal.
On most modern aircraft the oxygen is produced chemically in a canister, operated by the pull of a lanyard which releases the operating mechanism and fires trhe cartridge.
As one canister willl normally supply 3 or 4 masks, then the mask hanging down next to you in the vacant seat will (if its fed off the same canister) be releasing oxygen as well. it only needs 1 mask to be pulled for the others in the group to operate.
Depends on the system and the aircraft. From the user perspective, ususualy pulling the pin attached to the mask lanyard, ("pull the mask toward you") only allows oxygen to flow to that mask. The pin allows the flow valve to open to that mask only.
Oxygen flow typically opens the mask doors and drops the masks, and oxygen is flowing to the masks through a manifold (depending on the aircraft type and system) based on a pressure switch's response to cabin pressure altitude. When the cabin altitude reaches a preset value (typically 10,000' to 14,000'), the masks drop...because oxygen is being provided to them. Alternately, the crew has the capability of deploying the masks.
Some systems may use a gang valve system at the common mask attach manifold, some don't. The method of delivering the oxygen, acuating the system, etc, is not universal and varies with aircraft type and manufacturer. The differences aren't significant to the user. From a passenger perspective, if you get on the airplane and the mask drops, you grab it, pull it toward you to pull the pin in the manifold, and begin breathing. The differences are significant to the maintenance personnel who work on the system, and generally only then if they work on more than one system.
Not all systems use chemical generators, either.