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The transformer ratings are definitely non-critical. As this circuit uses the flyback topology, the ferrite cores must have a tiny air gap between them.You could use any small ferrite transformer for this. It's the turns ratio that's critical. A good way to estimate it would be to divide the output voltage required by the input voltage. For fast charging I would recommend keeping the turns ratio around 1:10.I'd highly recommend trying to wind your own transformer, it's easier to keep everything under control and definitely rewarding!All the best!
Yep, you can. Usually carbon or graphite are better than metals since they do not 'stick' to the rails.