On the Fire Planet
You jump from the platform you're on to the rocks floating on lava geysers. When you get to the last jump (5th to 6th rock), you jump when both rocks are moving downward. It will fall under you. Then you jump up onto the volcano and enter it.
Inside the volcano, you have to hide in the nooks when the hot air blows, and at the end jump over the "eye snake" once it fully unrolls.
(see related question)
You do not get or use a bucket on Astro Knights Island.
There is no treasure chest on Astro Knights Island.
you jump over it
There are no bird eggs in astro knights island.
Sir peleas gives it to you at the bottom of the volcano cave. (see related question)
No mines. But there is an underground chamber in Arturus. And the volcano passages are underground. (see related questions)
its on the island
How do you talk to the girl in the hay loft at astro knights island?
you just jump to the next one when you get near it!
On every island you have to be the one to save whatever needs saving. That includes the kingdom in Astro Knights Island.
On the Fire Planet, you get it from Sir Peleas when you reach the bottom of the interior of the volcano. (see related question)
Once you have built a new spaceship on Astro Knights Island, it will park on top of the castle whenever you leave the island.