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Here in Australia (NSW specifically) the highest level of high school maths in Year 12 has a topic on the logic and methods of proof. This includes general concepts of proof (symbolic logic, truth tables, the contrapositive, proof by contradiction, proof be counterexample, etc.) and some specific methods of proof. I think the best way to get a feel for what ...

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You can try the Rogers–Ramanujan identities: The number of partitions of $n$ in which adjacent parts are at least 2 apart is the same as the number of partitions of $n$ in which each part ends with 1,4,6,9. The number of partitions of $n$ without 1 in which adjacent parts are at least 2 apart is the same as the number of partitions of $n$ in which each part ...

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Your list seems like overkill to me. As far as geometry, Kiselev is basically a rehash of Euclid, so I don't see the point in studying both. Just pick one. I don't think you need the solid geometry parts of either. If using Euclid: -- Euclid contains stuff like number theory done in an ancient style that is now only of historical interest, so if using Euclid,...

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