Get the map (vent blueprint) from the window near the roof, and enter the Carrot Cake Factory by using the crowbar on the sewer pipe. One you are past the trap doors and the rat, enter the factory floor and press the three red switches at the left : the middle one all the way down and the other two halfway down. This will start up the machines. Climb right, then to the upper left, and ride the large crane arm across to the vent entrance at the upper right.
That is not part of the 24 Carrot Island quest.
This was designed as a fast way to go back outside the factory if you did not yet have the blueprints for the vent system. At the end of the game, you leave using a rope from the Launch Room, so there was no need to use it then. The sole value of the Transporter to the game is to escape from a game glitch, or to rapidly leave the island from somewhere in the factory. This was designed as part of the Haunted House game as well -- if you get stuck and have the Haunted House card, it lets you exit to the Island Map.
The game "24 Carrot Island" is the 3rd island introduced on the interactive website called Poptropica. On each island, the player finds a problem that needs to be solved for the island inhabitants. There are often mini-games as part of the solutions, and interactive competition with other users in Main Street gathering places. On 24 Carrot Island, the carrots are disappearing before they can be harvested. The island depends on carrots for their economy, including the now-closed Carrot Cake Factory. There have also been mysterious disappearances of local residents. The player must determine the cause by investigation, while somehow stopping the nefarious enterprise. (see related question and link)
24 carrot is a type of gold
You can't. At the start of the game, the carrots are being pulled down by an underground machine, part of Dr. Hare's evil plot to control the world. Finding and stopping him are the goal on the island. Start by catching the missing cat, so you can get into the abandoned factory.
Nothing. It is not part of the quest. The reason that Dr. Hare is stealing all that carrots is that he uses the Carrot Blender to turn them into rocket fuel for his Giant Space Rabbot. (really)
You eat the root of the carrot plant. (The orange part) You can also eat the top of the carrot, but I do not recommend it.
They are called carrot tops, leaves or carrot greens.
The elevator at the factory in poptropica is located in the far left part of the factory. When you arrive at the island you must go left to reach the factory passing the Robo-Bling boutique, which has a toolkit that you can use.
The black cat on Spy Island is just decorative and not part of the game. The gray cat, Whiskers, on 24 carrot island, can get you a crowbar. (see related question)
The entire carrot. The leafy bit, the main orange bit, everything... it wouldn't be a carrot else ;)
The root of the carrot is what is eaten.