Displaying the data export/feed icon only when there are search results
You can override the template for the views. 

There is a variable called $empty this field contains the text to be displayed if the node is empty. You can use this to check if view is empty or not. 

If you have placed the attachment before the node then you would edit the below section.

&lt;?php if ($attachment_before): ?&gt;
    &lt;div class="attachment attachment-before"&gt;
      &lt;?php print $attachment_before; ?&gt;
    &lt;/div&gt;
  &lt;?php endif; ?&gt;


You can change this to 

&lt;?php if ($attachment_before &amp;&amp; !$empty): ?&gt;
    &lt;div class="attachment attachment-before"&gt;
      &lt;?php print $attachment_before; ?&gt;
    &lt;/div&gt;
  &lt;?php endif; ?&gt;


This would make sure that the attachment is printed only if the view has a result.