When a city or county in Texas builds a new road or substantially changes an existing one, for instance changes a 2 lane road to a 4 or 6 lane road, does the state require them to do a speed study and set the speed limit using that study? Or can they simply just decide an arbitrary speed limit and post it?
My understanding is that if a state or local government builds or maintains roads using federal funds, that it is required to follow a variety of U.S. Department of Transportation design standards in connection with the roads built or maintained with those funds, and that the USDT audits its grant recipients for compliance with those design standards. The "85th percentile rule" that you allude to, as it is defined by the U.S. Department of Transportation in its regulations, is among those design standards.
The Texas Department of Transportation may choose for simplicity's sake to apply USDT design standards to all roads in its jurisdiction, even if they are state funded rather than federally funded, but that would be up to Texas officials to decide.
Likewise, local governments might choose to follow USDT or Texas state design standards for road that are entirely built and maintained without state and federal funds for convenience (and also subject to some state regulations under the traffic laws, e.g., re how traffic signs are designed and placed, certain maximum speeds on certain classes of roads, etc.) but probably have more autonomy with regard to those road than with regard to federal or state funded roads.