Hello,

In order to remove hover styles you need duplicate the same styles to both :link and :hover.

Here is the code to hide icons whishlist and My account

.menu-item-wishlist .woodmart-wishlist-info-widget .wishlist-icon{
display:none!important;
}
.menu-item-my-account a:before{
display:none!important;
}

As for mobile menu. Navigate to Header builder, switch to the mobile view and add the category menu into the mobile menu element.

Best Regards