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Somebody born on a reservation is a U.S. citizen and can vote in U.S. elections. Generally speaking, a Federal Indian Reservation is land owned by the U.S. Government and held in trust for the tribe. They are not "Free and Independent States" to borrow a phrase from the U.S. Constitution. They are sovereign only to the extent established by Treaty and Federal law.

Oliphant v. Suquamish Indian Tribe, 435 U.S. 191 (1978) limits the ability of tribes to try and punish non-Indians. Under this decision, they generally do not have jurisdiction to prosecute non-Indians. However, the state and federal governments have limited, and concurrent, criminal jurisdiction with the tribes. So as a non-Indian you can be prosecuted for violating state or federal law on a reservation. For example, if you're stopped for speeding on a highway through a reservation, the tribal police can't ticket you but they can detain you while you wait for a state trooper to ticket you.

Be polite and you'll probably just get a warning for minor violations. Be a jerk or commit a major offense, and they'll spend the time waiting for a state trooper to deal with you.