Ok, I think I nailed it.
First lesson: Don’t touch the TinyMCE settings switching from. docrelative to rootrelative. You cannot use internal urls like [~999~] anymore in the TinyMCE Link-popup-screen. So, I put that back to docrelative.
Second: I used the base tag at the top of my template. This solves the problems and everything is working good now.
<base href="[(site_url)]" />
I changed my save_content.processor.php as David suggested and everything is working ok.
Maybe the change can be put in 0.9.6.4?