AviK
Military law applies on the base for members of the military. They subject themselves to it when they are sworn onto service. Military law doesn’t apply to local civilian employees on the base. They are subject to local laws and courts.
The military mail APO/FPO addresses are for the convenience of the military, not the foreign government.
An analogy might be made with Indian Reservations here in the US. Having 2 major casinos on Indian land (Foxwoods and Mohegan Sun) right here in Connecticut the topic comes up often. If you slip and fall while visiting the casino you have to sue in the tribal courts, not state courts. Your entry is considered consent to jurisdiction. Employees (mostly non-tribal members) are not covered by state wage and labor laws. However the land is still part of the sovereign United States, not the tribe. The tribes cannot erect customs and immigration checkpoints and/or negotiate treaties with other nations.
AS for standing, I do not pretend to know enough Illinois state law to expound on their requirements for standing. They may be far different than the federal court requirements. Without reading the suit one does not know the exact claim being made to exclude Cruz from the primary ballot. This case has nothing to do with the national election in November.