The solenoid doesn't have anything to do with getting power to the sparkplug. The solenoid is just a relay that sends battery power to the starter when the start button is pushed.
Yes, the stator is similar to an alternator. The stator is actually a generator that generates power for the charging system. Since a generator generates AC voltage, the power is sent to a regulator/ rectifier where the power is rectified to DC and regulated to the proper voltage.