Custom Theme Fields in Settings Menu – apply filters to one of those fields?

You can create a custom filter using filter functions of Plugin API.
I have changed you code as following to create and use custom filter.

function add_gcf_interface() {
add_options_page('Other', 'Other', '8', 'functions', 'otherGlobalOptions');
}

function otherGlobalOptions() {
?>
<div class="wrap">
<h2>Sonstiges</h2>
<form method="post" action="options.php">
    <?php wp_nonce_field('update-options') ?>
    <p><strong>Welcome Message</strong><br />
    <textarea name="welcomemessage" cols="100%" rows="7"><?php echo apply_filters('welcomemessage',get_option('welcomemessage')); ?></textarea></p>
    <p><input type="submit" name="Submit" value="Save" /></p>
    <input type="hidden" name="action" value="update" />
    <input type="hidden" name="page_options" value="welcomemessage" />
</form>
</div>
<?php
}
// Create custom filter
function add_welcomemessage($welcomemessage) {   
return $welcomemessage;
}
add_filter('welcomemessage', 'add_welcomemessage');

Tell me whether it worked for you or i will provide another solution.