I agree with Steve. You could pick whatever flat bottom state badge comes for sale first. Without a sweat shield, and yours is missing, there it is little chance that we could identify a specific state police force. As for time period, there are endless variations of these police hats and I wouldn't even attempt to guess with any certainty. Although I would say earlier rather than later.
Neither - it is wool winter cap of Zivilbevölkerungsschutz (civil defence) or ZB. These caps were blue-grey with altgold buttons and worn without insignia, so nothing misses. Issued from mid 60s onward, yours appear to date 1965.
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