How can I jump to the next search cscope result in vim and open it in the currently active split? If I open :cw, the selected result will always open in the top-right split, which is not what I'm looking for.
:cwindow will open the quickfix list in a window (quickfix window) if there are results (Similar to :copen). You do not need to use the quickfix window at all in your case. You can simply go to the next quickfix item via :cnext.
For more help see:
:h quickfix
:h :cnext
:h :cw
:h :cprev
:h :cl
:h :cc