Does anyone have the solution for puzzle 6-7? :/ I can't manage somehow…

This is the same as puzzle mode number 13 (1-13)

I got hung up on this because I didn't realize that you could swap the current ball with the next ball!

Check out this youtube vid that explains how to do it (it starts at 8 and goes to 14, so just skip the first few.

Link: http://www.youtube.com/watch?v=sfXuKGksPNs

How can I find out to swap the balls?

right click your mouse...that will swap the active sphere in your winged shooter for the one "on deck."