How can I keep my last search string in the "Text to find"-textfield? When I search for something and move the cursor to another point in the document, I get the word my cursor is on when I type Ctrl + F. How can I disable this "feature"?

Try clicking on the dropdown box in the 'text to find' field - in there, you'll find all past search words with the most recent at the top.

Yeah, I already know this. But I'd like to have a way, where I don't have to waiste a click for this. Anyone an idea?

IF you want to continue searching (or relacing whatever you did last time), just press F3 and it will go forward or Shift-F3 and it will go in the opposite direction.
IF you need to open Search (or search/replace) Dialog and do not want a word under cursor picked up from the editor, just uncheck Find Text Under Cursor in the editor settings.

That's what I was searching for - thx !