Why some links don't have the domain name
e.x. design/
other links are with the normal way www.myname.com/host/

Check the source code of the links, maybe they are simply not created correctly. (absolute vs. relative links)

How can I check this?
If I will deactivate (not uninstall) the JoomSEF I can see the same error again.

Then I am most likely right. (see the source code of the article, menu item or whatever where the link comes from; all internal site links should either start with "index.php?..." then they are processed by JoomSEF or other router, or they should be relative to site root, /xxx/xxx, or they need to be absolute www.site.com/xxx)

The source code is like this <a href="//hosting">
Somethink going wrong I think.

Guess so, maybe your WYSIWYG has bug or is not configured properly. But I guess that is already out of range of these forums.