My first problem is not to be able to delete the "Blog" -looks-like-a-button but i guess is a page- on the left lower corner of the K2 Lite theme. It overlaps my customized header.

A custom menu should help with that. You can create a custom menu using the Appearance → Menus page in your blog dashboard. Make sure you set the "Primary Navigation" option on that page. I checked your Appearance → Menus page, and it looks like you created four separate custom menus but you didn't set any of them as primary.

Here is a help page with details and visual examples showing how custom menus work:
http://en.support.wordpress.com/menus/

If I add a new page, a button-menu will appear next to it which i can delete but i cannot delete this one blog which cam with the theme. It was containing the default hello world! post. So i start to believe that in this theme maybe i cannot modify that. I searched in the forums but without luck.

I am ready to migrate to another theme or create my own theme if necessary to solve this. I do not find the option to start a theme from scracht and then shift my blog to it.

Thanks for the help