Hi William,
To make the custom menu only available to those who are logged it is easy enough to do. You drop the reference for the custom theme from the frontpage.php file.
This is the code as found in base theme ...
<?php if ($hascustommenu) { ?>
<div id="custommenu"><?php echo $custommenu; ?></div>
<?php } ?>
</div>
<!-- END OF HEADER -->
And to answer the 2nd part of your question...this is Theme specific.
An example is Decaf theme.
Mary