May an adversary be fooled by a random-looking input that in reality is fixed?
Would an adversary realize that it is not interacting with the scheme but with a &quot;simulation&quot; if it finds out that k=5 in the challenge setting?

This is commonly stated as Kerckhoff's Principal, which I prefer to cite as Shannon's Maxim (they both stated it, and Shannon's Maxim sounds better). The idea is, the adversary should be assumed to know everything about a secure system, except the key, if it is to be considered secure at all. As such, the adversary must be assumed to know that you could lie to them about the value of $k$.
Also, a signature scheme could not be considered secure if it involves an oracle capable of lying about what signatures are valid or invalid. As such, if the system is to be secure at all, it must behave the same for an attacker as if they were a legitimate verifier.