Since I am new here also I will answer this question, you should realise though that I havent even been here long enough to install esyndicat so there may be issues I am unaware of.
The search box is just a form, as long as you use an URL that resolves to your search.php in the action="" part of the form, you can put that form anywhere, in any page, any site etc.
For example, from a view source this is copied strait from the demo at esyndicate.com/demo - notice the action="" part uses an absolute url;
<div class="search-form">
<form action="http://www.esyndicat.com/demo/search.php" method="get">
<table cellpadding="0" cellspacing="0" align="center" style="margin-top: 20px;">
<tr>
<td rowspan="2" align="left">
<ul class="search">
<li><a href="http://www.esyndicat.com/demo/search.php">Advanced Search</a></li>
</ul>
</td>
<td style="color: #FFF; size:14px; text-transform: uppercase;">Search</td>
<td colspan="2" valign="middle" style="padding: 0 5px 1px 7px;"><input type="text" class="what" name="what" size="40" /></td>
<td colspan="2"><input type="image" src="http://www.esyndicat.com/demo/templates/GreenLeaves/img/btn-search.gif" name="search_top" value="GO" /></td>
<td>
</td>
</tr>
</table>
</form>
</div>
Open up a new file in Notepad/TextEdit or similar, copy and past the code and save it to your desktop with .htm exstenstion, save it, then open it up in a broswer, type in a seach term and viola...
Now, as for adding featured links to other pages, that could be a little bit more tricky, maybe use RSS or you'll need to learn a bit of php and mysql... as I said I havent even looked at one line of code in this sfotware... yet.