If I remember correctly, the story about the new senator Ed Markey is that when he was in Congress he had an apartment in DC and used his mother's address in Massachusetts to register in his district.
Different states have different laws, though. And the laws may not even matter. See the recent Rahm Emanuel case. He rented out his Chicago place and moved to DC, but used the address to register to vote in Illinois, and eventually to run for mayor of Chicago.
My guess is there won't be any legal problem with Cheney running. It's more something emotional voters will have to sort out themselves.
One can also be a US citizen and be a resident of commonwealths, territories and possible a foreign country without being a resident of a State.
I'd like to understand exactly what the rules/laws are.