Talk to Sir Gawain and he'll give you the Force Shield. When the copter drops round snowballs, bounce them back at him, aiming for the yellow lights. Each time he's hit, he'll crash and shoot at you. Jump. After three hits, he's defeated.
If your shield gets depleted, Dodge the ice missiles until it recharges.
You can't. You must defeat all three planets, getting all three knights with you, before you can save the Princess. To defeat the Tigercopter, use your force shield to bounce his round snowballs back at him, aiming for the area with the yellow lights. When you hit him, he will crash and shoot at you. After three hits, he is defeated.
You get the Force Shield (one of the Mystical Weapons) from Sir Gawain on the Ice Planet (X-73 Y-83) where you use it to defeat the Tigercopter. (see related question)
You must defeat all three planets (tigercopter, phoenix birds, and dragon) to advance to the end of the game. When all three knights are accompanying you, you can find the Princess using the Golden Key on the secret giant asteroid. (see related question)
It is generally referred to as the TigerCopter or the Snow Panther Copter (it has the face and ears of the Pink Panther, only white).
You get the force field from Sir Gawain on the Ice Planet. You use it to bounce back the snowballs at the attacking boss, the Tigercopter.
No. (see related question)
When you defeat Mordred, the game is over, and the Princess will kayo him for you. (see the related question)
You do not get or use a bucket on Astro Knights Island.
You don't defeat knights, you talk to them and they'll give you a weapon to defeat a beast. (On the jungle planet, find the Laser Lance down in the plants before you meet Sir Cador.)
There is no treasure chest on Astro Knights Island.
The Princess isn't there. Sir Gawain will accompany you as you complete the other two planets. Once all three are done, you get a clue to the location of the Binary Bard's hideout.
If you retreat from a boss (the dragon, phoenix, or tigercopter), you cannot complete the planet and therefore cannot win the quest. (see the related questions)