Regarding the rights, the TOS has this to say:
When purchasing a Gig, Buyers are granted all rights for the delivered work, unless otherwise specified by the Seller on their Gig page. Note: some Gigs charge additional payments (through Gig Extras) for Commercial Use License.
and also this section:
Ownership and limitations: When purchasing a Gig on Fiverr, unless clearly stated otherwise on the Seller’s Gig page/description, when the work is delivered, and subject to payment, the Buyer is granted all intellectual property rights, including but not limited to, copyrights for the work delivered from the Seller, and the Seller waives any and all moral rights therein. The delivered work shall be considered work-for-hire under the U.S. Copyright Act. In the event the delivered work does not meet the requirements of work-for-hire or when US Copyright Act does not apply, the Seller expressly agrees to assign to Buyer the copyright in the delivered work. All transfer and assignment of intellectual property to Buyer shall be subject to full payment for the Gig and the delivery may not be used if payment is cancelled for any reason. For removal of doubt, in custom created work (such as art work, design work, report generation etc.), the delivered work shall be the exclusive property of Buyer, and Seller assigns all rights, title and interest in the delivered work. Some Gigs (including for custom created work) charge additional payments (through Gig Extras) for a Commercial Use License. This means that if you purchase the Gig for personal use, you will own all rights you require for such use, and will not need the Commercial Use License. If you intend to use it for any charge or other consideration, or for any purpose that is directly or indirectly in connection with any business, or other undertaking intended for profit, you will need to buy the Commercial Use License through a Gig Extra and will have broader rights that cover your business use.
Long story short, tell them no, direct them to the Terms of Service if they get sassy.