When the Sheriff puts the warrant on the dash it is folded in thirds, but when they arrive at the house it's folded in half.
The warrant says "United States District Court" at the top. A state policeman would seek a state court warrant, and a federal warrant would be served by a federal agent, not a state policeman.
After Chiana climbs into young John Crichton's truck, she grabs the radio tuning knob which turns up the volume...
The radio mentions the US invasion of Nicaragua which took place in 1987. This episodes takes place near Halloween, (end of October) 1985.