This is call for urgent help, please.

But I have added new child portal and set my skin to it.

Now to the problem:

When I am logged in (as host) my skin on child portal works fine.

But when I logout my skin mess from some reason. And I have found what is the problem but have no idea why it happens and how to solve it.

Inside skin I have this lines of code:

<dnn:DnnCssInclude runat="server" FilePath="zurb/css/foundation.min.css" PathNameAlias="SkinPath" />

<dnn:DnnCssInclude runat="server" FilePath="/skin.css" PathNameAlias="SkinPath" />

When I am logged in skin.css is loaded after foundation.min.css but when I am logged out skin.css is loaded first and result of that is my styles that are in skin.css are pverwriten by other css files.

Can somebody explaint me why this happens and how can prevent this and always load  skin.css after all others?

 Skin.css is loaded by default by DNN, there is no real reason to use an include like this.

I actually think you are loading it twice (once auto-loaded by DNN and once with the include).
I guess on the child portal the file is not found because the /Skin.css path references the path relative to the parent portal.

Remove the skin.css skin inclusion.