I tested in the default style and it works, just remember to change the css "margin-top" value to fit your need
in navbar template: search for
<ul class="navbar_advanced_search">
<li><a href="search.php{vb:raw session.sessionurl_q}" accesskey="4">{vb:rawphrase advanced_search}</a></li>
</ul>
Remove it.
Find
<span class="buttoncontainer"><span><input type="image" class="searchbutton" src="{vb:stylevar imgdir_button}/search<vb:if condition="$stylevar['textdirection'] == 'rtl'">_rtl</vb:if>.<vb:if condition="(is_browser('ie') AND !is_browser('ie', 7) AND !is_browser('ie', 8))">gif<vb:else />png</vb:if>" name="submit" onclick="document.getElementById('navbar_search').submit;" tabindex="100"/></span></span>
change it to
<span class="buttoncontainer"><span><input type="image" class="searchbutton" src="{vb:stylevar imgdir_button}/search<vb:if condition="$stylevar['textdirection'] == 'rtl'">_rtl</vb:if>.<vb:if condition="(is_browser('ie') AND !is_browser('ie', 7) AND !is_browser('ie', 8))">gif<vb:else />png</vb:if>" name="submit" onclick="document.getElementById('navbar_search').submit;" tabindex="100"/></span></span>
<div class="navbar_advanced_search" style="position:absolute; top:7px; left:100px; width:100%;"><a href="search.php" accesskey="4">Advanced Search</a></div>
Add the following css to your additional.css
This for moving top links to under navbar
.toplinks {
color: #FFFFFF;
font: 11px Arial,Tahoma,Calibri,Verdana,Geneva,sans-serif;
margin-top: 85px;
position: absolute;
right: 20px;
text-align: right;
top: 0;
z-index: 1;
}
This one for moving advanced navbar search
.globalsearch {
width: 290px;
}
For the facebook connect , check out this articles by Lynne
https://www.vbulletin.com/forum/showthread.php/361140-Moving-Facebook-Connect-out-of-the-header-template-and-into-the-navbar-template
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Added some changes, I think that's all, hope I didn't miss anything.