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In Poisson and negative binomial regression, the response is assumed have Poisson and negative binomial distributions respectively. When we test the significance of the parameter $\beta$, which is estimated by maximum likelihood estimation, we have the Wald chi-squared statistic
$\left(\frac{\beta}{\text{standard error}(\beta)}\right)^2$.
The questions are:
The results are more general than the ones asked of this question. The tests are generally used for most Generalized Linear Models (your specification choses two particular link functions i.e., functions describing the connection between the mean of a distribution to a predictor variable). The proofs for the most general case can be found in any standard text on large sample theory (see for instance Pg 508-510 in 'Testing Statistical Hypothesis' by Lehmann and Romano).
