About that button that allows a user to change the quantity of the item in the shopping cart. It seems to be called the buttonUpdate when we're talking about a product's attributes; its constant is ICON_IMAGE_UPDATE.

I got rid of it in the shopping cart by commenting out two separate lines (as it was appearing twice) in /catalog/includes/templates/template_default/templates/tpl_shopping_cart_default.php (after I made the changes, I saved it to /catalog/includes/templates/custom/templates/tpl_shopping_cart_default.php where custom is my custom folder). The first is around line 80 ( echo $product['buttonUpdate']; ) and the second is around line 143 ( <div class="buttonRow back"><?php echo zen_image_submit(ICON_IMAGE_UPDATE, ICON_UPDATE_ALT); ?></div>
)


You can get rid of the "estimate shipping" button from the configuration in admin. Go to Configuration => Shipping/Packaging => and setting "Shipping Estimator Display Settings for Shopping Cart" to 0.