You would be better off using 8 magnets and 6 coils, for a 3 phase generator.
I am not exactly clear on the physical plan.
It sounds like you want to make a single rotor generator. If so, I don't expect you can get enough turns of #16 close enough to the coils to make much power at all.
While #22 would work, the resistance would be very high, making the output low and inefficient. And I sort of doubt you can get the wire out of a wall-wart and still have it be usable.
The next problem is the 4" dia metal plate. That's awful small. Tend to run out of room in the center.
No doubt it can be made to make 12V with the parts listed, but no way to make enough electrical power to be worth running a lawnmower engine.
Ben's project would be a good thing to look at.
Follow the links backwards through the evolution.
http://www.fieldlines.com/story/2009/2/26/213016/068
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