Are you using Replete or this is custom merge of the two themes that you spoke about on this thread http://www.kriesi.at/support/topic/big-slider-from-incarnation ? By that I mean, are there parts from multiple themes you bought inside Replete, or is it Replete without 'extra parts'?
1) - Cart dropdown displays nothing: -
Have you create a woocommerce directory within your theme to override the plugin?
Please look at file mini.php in woocommerce plugin folder in /templates/cart/ directory. Does line 50 look like this
<li class="empty"><?php _e('No products in the cart.', 'woocommerce'); ?></li>
Have you modified the templates themselves or the theme files? I would suggest to download http://www.winmerge.com (or whatever equivalent if you have mac/linux) and to download a clean folder with the woocommerce and with the theme, and do a folder to folder comparison. You should instantly see where the differences in the code are in every file on a line by line - side by side basis.
2) - Checkout page is half screen
Have you investigated if this is propulsion css?
3) - Icon Insertion
*I will add something here shortly , working on it.*