I would never make that assumption. I would request that the seller purchase insurance and I would of course reimburse them for that expense. Same as I reimburse for the cost of shipping.
Not if you arranged the shipping. If I paid you money to deliver me a product, then it is up to you to choose the shipper and arrange the shipping and make sure it gets to me. If it doesn't, it's your responsibility to make it right. I does not matter to me what third party you may have chosen to use to ship the product. That's between you and them. If I paid you, then you are responsible.
If those were your terms of sale, I would not choose to purchase from you. The reason is it can take weeks or months to get resolution from a shipper to pay out an insurance claim, if they pay at all. If I paid the seller to ship something to me, then I paid the seller to ship something to me. I did not pay UPS or USPS or whoever.
Again, I would not purchase under those terms as it is too risky to send money to an individual who won't make a commitment to deliver what they sold or give a refund if they are unable to deliver.