1 http://s017.radikal.ru/i408/1408/42/e8002f39f4ee.png
2 http://s019.radikal.ru/i626/1408/79/bd9e3ee05678.png
3 http://s019.radikal.ru/i615/1408/8e/031875f84d79.png
please see pictures
using page up/page down does not work. I don't know how to pull this pink thing in order not to affect other paths

Without the svg file and a better explanation there are too many possible ways to interpret what you want and what might be stopping you.
Locked object?
group?
Don't worry, though! Two and a half of the moderators here are psychic and they will be able to figure it out when they show up. And bend your spoons!

Hey, I see the problem!

There is a "loop" between the object's ordering, that you want to achieve.
Hide the ear part with that purple path, hide the purple path by the outline object, then hide it partially with the ear.

To solve this, you will have to decide which object you want to keep on the top, and the z-order of the objects according to it.
Probably the most straightforward ordering would be to keep the ear part at the top.
Thus, you would need to clip it with a path, that shares a segment following the purple path's edge.

That purple path selected on the second screenshot could also be on top, but then it would need to be edited to follow the outline.


Another different solution would be, to represent that previous path with two different one:
one at the bottom -which is hidden by the outline- and one at the top -which can hide the ear-.
Probably this could be made the most clean on small details, though rendering issues in the others wouldn't show up that much either.

Edit: wasn't sure about the character, so wrote ear instead horn.

Wow, Daniel, you too? Three and a half psychics on the forum then.

When it comes to unicorns - all of my senses are sharpened

Joking aside, this is quite often situation in the preparation of characters. Hence this engaging.

"pull this pink thing"??