The Qixing each administrate the commerce of their own domains, as well as regulate the laws of Liyue as a whole. Civil disputes and mortal crimes are under their purview to resolve. Rex Lapis only intervenes when events involve a power that mortals cannot contend with.

[ Luckily, Teyvat hasn't progressed to the point of needing passports. ]

Yes, all are welcome to conduct business as long as they obey the regulations. Failure to follow the law, or breaking a contract, will result in punishment.