They will have invested their money to use a developer to build out extra required modifications.
1)
a) This is default, as per our demo:
b)
To hide the add to cart for mobile devices on the shop page, you can use this:
@media only screen and (max-width: 767px) {
li.product figcaption .shop-actions {
display: none;
}
}
To add a custom add to cart button in a different position for mobile would require you to work with a developer for this modification.
2) This would require you to work with a developer for this modification.
3) To change the cart icon for mobile, please install and activate the supplied child theme, inside the child theme functions.php
file paste this code: https://gist.github.com/webadelic/6d8eceb08f3d00671ed9a380471969e0
Locate the icon code: <i class="sf-cart"></i>
and replace with the icon of your desire.
4) You enable the option: Theme Options => Header Options => Cart aux item count