1. Yes
2. "ALL" income received from a client/customer "for any reason/purpose" is business income. That would include shipping costs paid to you by that client/customer, even though you turn right around and pay it to the shipper.
3. Look at it this was (because the IRS does). All money received from the customer "for any reason" is business income. All shipping expenses incurred by the business is a deductible business expense. Period. Shipping paid to you by the customer does not have to match what you pay the shipper. In fact, it rarely does for most businesses - as a small percentage of the business profit is the overcharge on shipping they pass to their customers. Typically a business will overcharge shipping to the customer anywhere from 5 to 25% so as to ensure the business does not incur a loss of profit on the product being shipped.
4. Shipping costs you pay are shipping costs. Doesn't matter if its a shipping cost you pay to have a product sent to you, or shipping cost you pay to send product to a client/customer. Generally, shipping costs for products that are sent to you are included on your books in the price you paid for that product. Whereas shipping costs you pay to send product to a client are included as a part of your COGS (Cost of Goods Sold).
The first is claimed/deductible in the tax year you pay it. The 2nd is claimed/deductible in the tax year you sell the product.
