Yes, when the flywheel is connected to the grid, the generator will act as a motor, drawing power from it to bring the flywheel up to speed.
Once the flywheel is up to speed, a tiny amount of power will be used to make up for the losses in friction caused by the bearings and air resistance.
No, the generated power will be AC since it's an AC motor/generator.
The generated voltage will only be very slightly lower than the nominal grid voltage due to the losses in the generator, possibly a volt or so? This makes very little difference because when the grid voltage drops below the generator's voltage, the energy from the flywheel will be put back into the grid.
No, there is no need for a converter circuit, all that's needed is a synchronous motor with a huge flywheel connected to its shaft. Having said this some circuitry may be required to limit the current surge created in starting the flywheel but it certainly isn't required once it's up and running or when energy is taken from it.
I imagine that putting the flywheel in a vacuum and supercooling the motor/generator will minimise the losses but it probably isn't worth it.