Zen header 'site name' + links issue
I am new to coding so anyone should feel free to correct are add to this.

Your theme is placing the content on a page using a file called page.tpl.php. It may be found in your subtheme folder or in the parent them folder. This file is responsible for assembling the HTML code that actually displays your page.  The file is a text file and is easy to open and edit (not as easy to understand if you are not familiar with HTML and PHP.  I am using Fusion themes, so there may be some difference, but if you search for 'menu' you will find the part of the template that generates the menu.  There you will probably find a HTML command to 'clear' or a Class definition that will include the clear command.  This command is telling the browser to push further content in the region, such as your block, to the next row.  To change this you will need to make an override page.tpl.php file (copy the parent theme file into your subtheme directory and remove the command. If the command is in the CSS file, you will have to alter the CSS file containing the class definition.  

I do not recommend doing this if you are not yet familiar with coding in Drupal (do a bit of research first).  Good luck.