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SPIN is not the obvious choice here - domain-specific tools dedicated to protocols are more suitable. Some examples can be found here: Tools for modelling and analysis of cryptographic protocols

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No. Usually the Kerckhoff principle is considered. That means that the security of a cipher should rely on the key alone. The algorithm itself should be considered known. You could think of the algorithm as a single key that anybody that knows the algorithm possesses. Such a key should not be considered secure. There is no generic way of testing the ...

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If Eve knows $C, \space N, \space K_{CA}(N)$ then she is able to impersonate $C$ just by following the protocol you described above. If she just knows $K_{CA}(N)$ for a single $N$ then the effectiveness of the attack depends on the set wherein $N$ is picked. But as you are supposing that the communication channel is not secure, an attacker could spy ...

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