A few random answers:
2: This is the Categories-Tabs menu, and you can turn it off in admin > Layout Settings.
3: You can move the header navbar in /includes/templates/your_template/common/tpl_header.php. Find the block of code enclosed between <!-- bof whatever --> and <!-- eof whatever --> and move the entire block to the location you desire - in this case, after <!-- eof branding display -->.
4 & 5: These will be CSS issues. You have undesired margins or padding applied to various elements. You need to use Firefox with the Web Developer extension and see what id's, classes etc. apply, and then find those in your stylesheet.css and change them. (Try the CSS > Edit CSS button in Web Developer - it lets you see the effects of edits instantly.)
6: You can do this in the stylesheet by finding the id or class that applies to each of those headers, and editing or inserting their declararions. Add
Code:
background-image: url(../images/your_image_path);
to the appropriate declarations.
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