The Dungeon Cell Back Gate
The "gate" is in the back of the dungeon cell where the robot invader is being held. That gate is opened by the switch you pull when entering the dungeon from the library. When you push through the dirt wall in Mordred's underground lab, the gate should be open.
The Crystal Gate
The portal to the other galaxy is on the planetoid at X-11 Y-82 in the Asteroid Belt. You can only pull out the Golden Key when you have beaten all three planets, because you need all three knights with you.
(see related questions)
The coordinates are X 11, Y 82
Once the Golden Key and the Green Orb are returned to Arturus, there is no way to open the crystal gate.
The switch in the dungeon opens the back gate to the cell. Later, you can enter that way through Mordred's underground laboratory. (see related question)
Push through the dirt wall. You should already have opened the back gate with the switch in the dungeon. Feed the mouse to Merlin. He can retrieve things for you.
There is no way into the cell from the library. Throwing the switch opens the "back gate" to the cell, which you can access from Mordred's underground laboratory. (see related question)
When you get the moldy cheese from the castle dungeon, throw the wall switch that opens the rear gate to the cell. Later, you can enter through the tunnel from Mordred's lab.
To draw the sword from the crystal stone, you must have defeated all three planets (jungle, ice, and fire) and have all three knights with you. (see the related questions)
Feed the mouse to Merlin, the mechanical owl, and he will follow you into the cell. Click Merlin, then the vial, and he will retrieve the fuel rod for you.
The small gate is in the BACK of the dungeon cell, and connects to the tunnel Mordred dug. By throwing the dungeon switch, you open the gate so you can come in that way. (see related question : getting back in the dungeon)
Go back down and pull it. You need the gate open to reach the cell. (see the related question)
you fly to a large asteroid in the asteroid belt and land there. that is the crystal gate when you climb to the top
The robot invader is held in the castle dungeon, but there is no way to get to it. Go around and enter through the tunnel in Mordred's secret underground laboaratory. (When you enter and get the cheese, pull the switch that opens the back gate to the cell.)