Can I coil directly onto the iron cores of a stator for a basic generator?
Q1 that should be fine.

Q2 You will get this it is called cogging and is normal in this type of generator you need to apply enough force to overcome this.

You also need to give consideration to completing the magnetic circuit both in the rotor and stator. This allows the lines of magnetic force to easily propogate between each magnet in the rotor and each core in the stator. In its simplest form this would be an iron centre for the shaft which each magnet is attached to and an iron ring joining all the coil cores on the outside.

The magnetic force lines will propagating through the air. This gives you less magnetic field strength where it matters around the coils. A low efficiency generator can be made to work with this. If you look at a a good professionally produced generator it will optimise both the electrical and magnetic circuit.