Can I carry around a copy, or does it have to be the original that the FCC sent me? Does anyone ever ask for it?
Myself, I carry a laminated original license in my wallet for when I go to Ham events, Ham clubs, Ham organizations/Associations. Most of these, unless you are using a club and/or event call, want/need proof of you being a licensed operator.
I keep laminated copies of my Ham ticket in places such as my public service ID holder (for when I'm working events) and in my Go-Kits.
I live in New England and while no paperwork is required to have radios in one's vehicle(s), when on the road up here, New York state, as well as some of the southern states I travel into and through do...
Better safe than sorry. I know I don't want to be issued a violation, or worse, have my gear "removed" from my vehicle, as one of my non-Ham friends had happen to his scanner while in New York state. While trying to stay on topic, for reference, check out this link on scanner laws as they relate to US states:
U.S. Scanner Laws