The NP code is the same as the NUC (Network Unlock Code). The phone is locked to the Virgin network and requires a code from them to use another sim from a different operator. You may also find your local market trader or independent phone shop can unlock it for about £10.
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Yes, prime factorization is not an NP-complete problem. It is in fact in the complexity class NP, but it is not known to be NP-complete.
No, "np" is not a countable noun.
Proving that a problem is NP-complete involves demonstrating that it is both in the NP complexity class and that it is at least as hard as any other problem in NP. This typically involves reducing a known NP-complete problem to the problem in question, showing that a solution to the problem in question can be used to solve the known NP-complete problem efficiently.
Yes, the clique problem is NP-complete.
Yes, the partition problem is NP-complete.