Help configuring a local owncloud server to be accessed from the web

In you configuration (with the new DocumentRoot), this line is not needed:
Alias / "/var/www/html/owncloud/"
To redirect all traffic to SSL by default, you can change the normal HTTP-Vhost to:

<VirtualHost *:80>
    ServerName mycloud.net
    ServerAlias www.mycloud.net
    Redirect permanent / https://mycloud.net/
</VirtualHost>

To make sure all resources are send via https (and no mixed content), you can add this to your config/config.php:

/**
         * When generating URLs, ownCloud attempts to detect whether the server is
         * accessed via ``https`` or ``http``. However, if ownCloud is behind a proxy
         * and the proxy handles the ``https`` calls, ownCloud would not know that
         * ``ssl`` is in use, which would result in incorrect URLs being generated.
         * Valid values are ``http`` and ``https``.
         */
        'overwriteprotocol' => 'https',

Reboot webserver, clean browser cache, try on a current firefox/chrome browser.