How to display product code everywhere the ...

Hi,

You will have to edit the layout files you're using of the "product" view via the menu Display->Views in order to add that.

The modification isn't really complex but you need to do it for the correct view file.
On your first screenshot, it's the file "listing_img_title" where you will find the line:
echo $this->row->product_name;
that you can replace by:
echo $this->row->product_name.' '.$this->row->product_code;