The problem is not that the PHP doesn’t execute. It does execute but you get fatal errors because you are loading your template files directly, which means they are loading without the WordPress framework that supplies necessary functions. View source of those pages and you can see error.
You should not be linking directly to that file at all, or directly to any of your template files– for example, http://elliottequineconnections.com/wp-content/themes/elliottequineconnections/horses-for-sale.php
That is not how WordPress templating works. You should have created pages from the backend– wp-admin->Pages– for your content. When you edit one of those pages, you can see the permalink right under the title. That is what you need to link to. It won’t look like the actual path to the file.
You can get the permalink for that pages by using get_permalink( $id ); where $id is the ID of the page, but that is going to be tedious. You can make things easier on yourself by using a custom menu. You would do that by registering a navigation menu in your theme’s functions.php and then adding code to display that menu elsewhere in the theme.
Following the Codex, you would put this in function.php:
register_nav_menus( array(
'pluginbuddy_mobile' => 'PluginBuddy Mobile Navigation Menu',
'footer_menu' => 'My Custom Footer Menu'
) );
And then in footer.php you would place:
$defaults = array(
'theme_location' => 'footer_menu'
);
wp_nav_menu( $defaults );
// and the same for the pluginbuddy_mobile menu
Untested but that is about it. You will end up with a menu on the back-end under Appearance that will let you construct your menu with a GUI.