one of my favorite cards...

Questioned Claim
Any phase: Each player that does not control a King or Queen character must choose a challenge type. Until the end of the phase,that player may only initiate challenges of that type.

If someone trait changes a character after the event has resolved, do they get to initiate more then one type of challenge? Is the event a static effect or is it a one time effect when checking for Kings and Queens?

The latter. It checks for Kings and Queens at the time of initiation, and providing it isn't cancelled, the effect becomes a fact of the game that lasts for as long as determined by the card (until the end of the phase in which it was played). If I used Copper Link to add the King trait to one of my characters during said phase, I would still not be able to initiate any challenges other than the type I had specified.