For count data with excessive number of zeros, there are two choices of models, zero inflated poisson and zero inflated negative binomial.
Q1: How does one make appropriate choice between the two from theoretical viewpoint?
To me, zero inflated negative binomial has an extra parameter for variance part which allows more flexible variance structure as compared to zero inflated poisson.
Q2: How do I know this behavior beforehand?
Q3: Suppose one is compelled to choose between ZIP and ZINB. Is a badly fitted model(ZIP) with significant parameter better than a good fit model(ZINB) with insignificant parameter? The improvement of fitting is dramatic. I am leaning towards ZINB as it explains my data quite well but insignificance is obtained in return.