How to equip items in Zelda: Skyward Sword

You're supposed to make it dizzy by rotating your sword in a circle (not a spin attack). Do it quickly, but not too quickly—if you go too fast, you'll start doing random swings, which won't work.

For the eyes that appear later in the temple, you need to have all of the eyes in a given room look at you at the same time in order to get past them. If not all of the eyes are open, they won't get dizzy and die no matter how fast you rotate your sword.

No, there's no left-handed mode, although there were lots of rumours. This IGN editor's note confirms that from Nintendo itself.

When sites started reporting our "news" we immediately started to have second thoughts. Was there a left-handed mode? We consulted our combined 50 pages of notes and realized we had made no such observations. That lack of evidence caused me to quickly send an inquiry to Nintendo for extra verification. However asking Nintendo a question can take a little while, particularly if it's something a bit off the radar or about a very specific game detail...

...Late, late last week I heard back from Nintendo that there was no so such mode.