Do all NP complete problem have polynomial verifier?
A problem is in NP if and only if it has a polynomial verifier – that's the definition of NP. All NP-complete problems are in NP – that's part of the definition – and so have a polynomial verifier.

Factoring is not a decision problem, so it neither belongs to NP nor doesn't belong to NP. You can turn Factoring into a decision problem in many ways, for example asking if the $i$th bit of the $j$th factor of $n$ is equal to $b$. In that case a witness will be the list of factors of $n$. We can check that the factors multiply to $n$, and that they are all prime (since primality testing is in polynomial time).

(We are using here the unique factorization theorem, stating that a positive integer can be factored as a product of primes in a unique way. The same property doesn't hold in some more general situations.)

It is suspected that the decision problem corresponding to Factoring is not NP-complete, though it is certainly in NP, as the preceding paragraph shows.