ZKIP for Paillier public key correctness
First, the awkwardness of ZKPs and distributed key generation in Paillier is the reason many protocols use exponential Elgamal for protocols like mental poker. It is additively homomorphic but has the drawback on only being able to decrypt a small message space. In your case, since users are opening their encryptions, it seems suitable. The whole protocol could be run under a common key that is shared among the users (in a distributed or threshold manner).

I don't know of a ZKP for well-formed keys in Paillier. This is just an initial thought. In the following Paillier variant (from Encyclopedia of Cryptography and Security):



It should suffice to prove $n$ is a well-formed RSA number (e.g., CM99) and prove knowledge of $\lambda$ given $y=(g^n)^\lambda$ for $y=1$ (e.g., knowledge of discrete logarithm)?