You can click on the door behind "her", but if you give her the weapons, you have to solve the door puzzle to enter the final battle against Mordred.
(see the related question for the solution)
With Mordred
Solve the door puzzle and battle Mordred to save the (real) Princess. (see related questions)
I think that it was Mordred.
The Princess has been captured before the start of the quest on Astro Knights island, and is not seen until the final battle in the lair of the Binary Bard.
You do not get or use a bucket on Astro Knights Island.
No, she is not bald. She just has her hair in a bun under her crown.
The Princess Is Held By Alien's.
There is no treasure chest on Astro Knights Island.
She isn't on a planet.she is on a asteroid (if you are talking about astro-knights island)
No, because you need the three knights to go with you. (see related question)
The solution to Astro Knights Island is at the related question below. Your quest is to rescue the Princess, who has been kidnapped by robot invaders and taken into space. Find and repair the saucer Excalibur and launch into space, to find the three planets where space knights are searching for the princess. Defeat the robot monsters on the planets using the Mystical Weapons of Arturus, then face the Binary Bard in his castle in another galaxy. (see related solution question)
No, but there is no advantage to keeping them. The Princess is a disguise, and if you are fooled, you can still enter the castle by solving the puzzle door. If you click the door, you enter the final battle immediately. (see related question)