Regarding Thumbshots, the powered by image is required to use their free service. See FAQ entry #9 here
http://www.thumbshots.com/Support/FAQ.aspx
The second issue.. first make sure you have Base URL and Base path set correctly in the admin configurations panel.
From what I can see, you should have
Base URL =
http://www.vakantiehuizengids.com
Base path = /links
If that is correct, then you may need to edit the includes/functions.php file a bit.
First, make a backup copy of /includes/functions.php
Looking through the draw_sites_function which starts around line # 1140 (likely different for your version) find mentions of the tracking file "out.php"
Since the actual out.php file sits in your /links directory, "./out.php" needs to be changed to "/links/out.php"
You should find 2 mentions of out.php, one for the site preview pic/link, and one for the text Site name link.