As to why it is adding extra whitespace, I think it is because you are inserting the text from database into div directly (which contains some white space in html code, which is removed when page is rendered).
One efficient solution would be to insert your content in
tag, probably like this:
<div class='big_wrapper'> <p id='editable'></p> </div>
In jQuery, (which you are using) you can implement this :
data: 'tekst=' +$.trim(div_inhoud)+ '&pagina=' +pagina+ '&editor=' +$.trim(editor),
or in sent_data.php, you can use TRIM mysql function in your update query.