Rational procedure of fitting GLM zero-inflated model and potential pitfalls I would like to ask if the following way of thinking is valid. Some context first, we have a response variable which is count and a few other explanatory variables and also one random effect variable. Also, the zero counts by far exceed the rest counts (zero inflation problem). Hence, I would like to fit an appropriate GLM model.
1-Step) I would like to choose between Poisson and Negative Binomial distribution. I found the second one more suitable because of overdispersion.
2-Step) I would like to fit a GLM model with the most significant variables. For doing that I fit the full model with the random effect included and start to discard variables based on their significance (p_value of t-test, I can use also AIC, BIC, etc.)
3-Step) When I have the fitted GLM with the most significant variables, I would like to check if a zero-inflated model would be useful (I already know that there is a zero-inflated problem, but I want to have some verification). In order to achieve that I would fit a zero-inflated model with the previous most significant variables and prove that it can predict much more accurately the observed zero counts.
The order of the following three steps is rational?? Also, are there any pitfalls that I have to take a closer look at??
 A: 
2-Step) I would like to fit a GLM model with the most significant variables. For doing that I fit the full model with the random effect included and start to discard variables based on their significance (p_value of t-test, I can use also AIC, BIC, etc.)

This type of stepwise procedure should be avoided as it can lead to bias. A much better approach is to be guided by the underlying theory of the data generating process, and expert knowledge.

3-Step) When I have the fitted GLM with the most significant variables, I would like to check if a zero-inflated model would be useful (I already know that there is a zero-inflated problem, but I want to have some verification). In order to achieve that I would fit a zero-inflated model with the previous most significant variables and prove that it can predict much more accurately the observed zero counts.

You can't "prove" anything with statistics. You have already determined that the negative binomial model is warranted due to over-dispersion, so I would advise leaving it there, but if you must then of course you can compare the models with predictive accuracy.
