After all possible combinations, the last one worked so I don't know if you have to go through all of the combinations to open the door or not. The combination I got was Upper Left, Lower Right, Lower Left, Upper Right.

The Dual Blade fell into the sea with Doom Island. To go below the surface into Doom Island, enter one of the Whirl Pool like circles. Go through the cave until you reach Doom Island fortress. The Dual Blade is here.

The Power Oil is being guarded by the Oil dragon in Oil cave. To get to Oil cave, enter one of the Whirlpools (eddys) and exit through the other Eddy. You will then find the entrance to Oil cave. Kill the dragon and you will get the Power Oil.

The middle switch should do it.

Here are three sets of eight Dragon Egg locations:

First eight dragon eggs:

• Old Cave, 2nd floor
• Ghost Cave
• Red Tower
• Cave west of Odel
• Grief Tower
• Linze
• Gayas Cave
• Ullip Island Cave

Second set of eight dragon eggs:

• Alekia Castle
• Belgen
• Cave west of Linze
• Cave southwest of Belgen
• Elba Island Cave
• Underwater near Soshette
• Oil Cave
• Northwest Tower

Third set of eight dragon eggs:

• East Cave (east from Trek)
• Sheran Castle
• Lyden
• Old Cave, 7th Floor
• East of Herat Cave
• North Cave
• Southwest Cave
• Glasden Tower