How to copy and paste normally?

Hi peppy,
Nope. It's a browser security feature. Firefox and Opera do it. IIRC webkit (Safari, chromium) Doesn't.
It's the browser which forbids pasting and not CKEditor. CKEditor just warns you not to try to do it and provides you with an input box to paste into instead. All other WYSIWYG editors which use an IFRAME tag like CKEditor do the same. It's browsers which don't support pasting into an IFRAME tag. Web browsers can be exploited using that very same technique, which is why most browsers forbid it. Without that dialog which pops up, nothing would happen normally.
You can still use the function key combination ctrl + v to paste on Linux and Windows systems. I suspect OSX is Apple key + v.

Thanks,
Zanpakutō

Hi Peppy,
because as already explained, CKEditor uses an IFRAME tag as the editing area. It is a browser problem. Not a CKEditor problem.

Thanks,
Zanpakutō

Hi Peppy,
using an IFRAME tag as the editing area is a core part of CKEditor's architecture. Without completely re-writing CKEditor to add the option of using a different tag, there is no way to change it. I guess that an IFRAME tag is used because it is easier to apply entire style sheets to one and thus create document independent styles inside the editor. Meaning that syles that are important inside the editor aren't modified by any other style sheets which are loaded into the document. It's very easy to set styles for elements using CSS, however it can sometimes be very difficult to remove or replace them. If you don't know exactly what they are.

I personally wouldn't have used an IFRAME tag but I don't develop CKEditor, so my opinion isn't important in such things.

As far as your request goes, the editor used on this forum is a BBCode markdown editor and not a WYSIWYG editor because what you see is not what you get. For example :

[img][/img]

[Bumping this]

Can anybody tell me the difference of pressing CTRL+V and pressing the paste-button in the toolbar? Does the browser really handle this differently? Because I don't get a "paste dialog" by pressing CTRL+V in my browser?

If you press Ctrl+V, the browser knows that you want to paste something, and inserts that into the editor.
If you click on the toolbar of CKEditor, that's an image with some javascript, that javascript tells the browser: "please, paste the contents of the clipboard here", but usually the browser will refuse that request because it can't be sure that you are really wanting to insert there the contents of your clipboard (as it might contain private data), or if the page was some phising attack that showed a "win 10 dollars" button and used your click to request the clipboard data.

Thanks for your clarification alfonsomi - didn't think about that