1. If the mean equals variance property of the Poisson distribution is hardly believable, using the empirical-type standard errors is better than using model-based standard errors?
2. Link function is a monotone and non-negative function?
I think these are both GLM questions?
1) I would guess true; I think by empirical standard errors you mean some kind of sandwich estimator of the variance (for example, the same estimator you would get from doing M-Estimation based on the log-likelihood of the Poisson without making any assumptions about the variance) as opposed to using something based on the fisher information?
2) False; the identity function is a very common link, and clearly it can be negative.