WordPress.org

How can I show a link on the homepage only?

Use a Conditional Tag like is_home() if you add the link into the textarea field by the form.

If I using a free theme, can i change the footer links?

Look if the author are allowed to change it. Someones don't like or not allowed it. If you using a free theme by wordpress repositary in most themes you can change the footer links with your own code. Replace this footer links with the function <?php do_shortcode('[wplerlm]'); ?> and add some links for every blog to show your own links on the footer of this theme.