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>
Another solution is to trim the text before inserting into db. You can do that either at javascript post stage, or right before inserting in mysql db.
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.