For RSA one can give a non-interactive zero-knowledge proof that RSA with parameters $(e,N)$ form a permutation and a proof of knowledge of the associated RSA secret key. For example, such a proof can be given using Poupard and Stern. I was wondering whether there exists a similar proof for Paillier. In a sense that to prove that the parameters are valid, and that one posses the associated Paillier secret key. Any ideas?
The Poupard-Stern protocol proves knowledge of the factorization. Knowing the secret key in the Paillier encryption scheme is equivalent to knowing the factorization, hence the exact same protocol can be used to prove knowledge of the Paillier secret key.
The second part of your question, proving that the parameter $N$ is well-formed, is much, much harder. It has been studied, for example in this paper, but remains quite inefficient, since one must prove that committed numbers are prime, which is done by demonstrating in zero-knowledge that they pass all steps of a probabilistic primality test (each such step involving exponentiations, hence each such proof requires proving exponential relations between committed value, which is again relatively inefficient).