Are there any requirements to post, carry, or have available an official paper copy of one's radio license in order to operate in the U.S.? Do any states require physical possession of a printed paper license for mobile operation?
No. The ULS is the official document.
Since February 2015 the FCC no longer routinely issues paper licenses, although a paper copy can be obtained by request. The paperless policy was published October 2014:
We deem the electronic version of an authorization stored in ULS or the ASR System as the official Commission document.
Furthermore, §97.5 describes the station license requirements:
(a) The station apparatus must be under the physical control of a person named in an amateur station license grant on the ULS consolidated license database or a person authorized for alien reciprocal operation by §97.107 of this part, before the station may transmit on any amateur service frequency [...]
It looks like the only requirement for license information in physical form are 97.213(d) for telecommand of an amateur station (license and label with contact information posted at the station location); and 97.215(b) for telecommand of model craft (label with contact information on the station transmitter).
Being able to show your license while out and about "doing radio" is probably a good idea in this day and age, though.
HTH de AC0CW