What are these green pigs with red mustaches

Just some (pretty) flotsam - they don't play any part other than being an obstacle.

(You'll find lots more of such things in later levels - only items like the crates of dynamite and balloons are "active".)

This web page explains how the Mighty Eagle works.

The salient points are:

• You will not receive any stars or points from using the Mighty Eagle - your performance is considered in relation to a completely separate 'Eagle Score'
• It's not a one-use item - once you purchase the Mighty Eagle you will have permanent access to it. However you can only use it once per hour on levels that you haven't beaten yet.

So you could purchase the Mighty Eagle and then use it (without receiving any points or stars and thus not cheating), and the icon would presumably not be available on that level, or any subsequent unlocked levels for the next hour.

Note - I haven't purchased it myself, but after reading about it there's a strong possibility I will!