The intercom, chains/pulley system and table in Jack's cell indicate that this was an operating/treatment room for large animals. It used to be flooded to provide an observation/recovery area for the animals after treatment.
Walt was possibly kept in the glass room, where Jack is held prisoner.
Walt's initials can be seen engraved on the wall behind Jack when he is sitting on the floor with his back against the wall.
The Hydra is depicted on the maps that Sayid stole from Rousseau, instead of being labeled The Hydra it was labeled Alex.
The Hydra is station "C4?" on the blast door map. The second island would account for the apparent disconnect of this station from the main island. References to "KEV-4?" as a zoological station and ursus maritimus are also noted in this general area.
The Blast Door Map shows location C3 on a small island separate from the Main Island. C3 is the Hydra Station.
Alex managed to use the intercom to communicate to Jack that she had unlocked his cell door in an attempt to free him.
The messages on the intercom were sent by The Others to intentionally confuse or misguide Jack.
The Intercom has shown qualities similar to those that have been recognized in the entity we call The Monster, primarily the fact that Jack's dead father, Christian has spoken to Jack through it. Jack has been "contacted" by people through it, telling him things that would be hard or even impossible for them to know (Alex knowing that the door was open, for instance). He has also been able to overhear parts of conversations not intended for his ears. And he has not to our knowlegde been able to communicate back to any of these persons. Also, in "Every Man for Himself", after the initial message from Christian, we get to overhear Tom talking about how the intercom is still malfunctioning after the discharge - hinting that The Others are not using it, since they think it's not working.
It is Alex's voice on the intercom that tells him to "Try the door."
The "Hydra" name refers to the fact that the station is comprised of multiple, independent but interconnected, operating units -- "heads" -- just like the mythological hydra has.
Highly unlikely, since "The Hydra" logo shows the animal named Hydra.
The "Hydra" logo could hint that there are multiple things going around there, hence the hydra with its multiple heads. This, in contrary to, for example, the Pearl, which function is only to observe.
Legend states that the Hydra protected Crater the Cup, the chalice tossed to the heavens by Apollo. Also note, the Others (decoy) village appears to sit in an ancient crater. (taken from a posting on thefuselage.com's message board)