A Sport-Pilot ENDORSEMENT is relatively easy, but you will be limited to SP privileges. Day, VFR, etc.
An endorsement. from a CFI. 2 to 20 hours of dual time needed, but about 10 hours will be typical.
For a PP Gyro RATING is covered in FAR 61.109(d) which states the requirement for 40 hours of flight time,
it doesn't specify that an applicant for a gyroplane rating must have 40 flight hours in a gyroplane.
You probably already have 40+ hours in FW aircraft.
Specifically, you must obtain the following flight experience in a gyroplane:
3 hours of cross-country training;
3 hours of night flight training including one cross-country flight more than 50 nm in total distance,
10 Night takeoffs and landings to a full stop;
10 hours of solo, which include three hours of cross-country flight more than 75 nm in distance and to three different airports,
3 takeoffs and landings at a towered airport.
And Three hours of flight-test preparation;
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A one Million dollar Liability policy will typically be about $1500 a year as stated.
Dual or single will make little difference.
If you lawn-dart either type into a house, The damages from the fire will be the same.
Collision and comprehensive seems to run about 10% of the declared value.
An $80,000 Gyro will run about $8,000 for a year of coverage.
You can add Comprehensive that covers the Airframe for Fire, Theft, other accidents as long as the Rotor is NOT Turning to a liability policy for another few hundred dollars.