First the magnet mounting question, yes, embedding cause flux lost and potential eddy current losses when running. I made finite element models to illustrate. This is a 12 pole, 12 coil simulation with a 1.5mm gap, 10x10x3mm neo magnets on a 100mm dia rotor.
You can see the flux concentration at the magnet edges compared to surface mounting
The flux distribution in the gap is reduced and is not as even across the face
This example shows that the gap flux (generated voltage) will only be reduced 15 to 20% by embedding the magnets 90% into the steel rotor. Flush mounting would likely cause a greater reduction. I typically use a pvc or wood jig to hold the magnets while bondingmagnets, using a 2-part epoxy with good shear strength. The magnets will always be strongly attracted to the rotor, so we only need to overcome shear and centrifugal forces. When bonding magnets, keep the bond line thin and always clean all surfaces with acetone (not alcohol) before applying adhesive.
If the magnet flux is generally constant I use a mild steel, for a stator and pole pieces, I'd use ductile iron or lamination.
Aluminum for pole pieces would change your generator into an eddy current brake. So no.
Using Neo magnets could improve the bicycle generator if the air gap is not too small. You would likely get higher voltage at the expense of increased torque requirements. Friction drive? = abrasion of sidewall
For coil cores, use ductile iron or ferrite
Hope that helps