Generalized Linear Model and fixed effects

I want to estimate a model with a fractional outcome variable following Papke & Wooldridge (2002). I run glm dep_var independent_var, family(binomial) link(logit) robust, right?

How can I add a three level fixed effects? My approach until now was to use egen new_var = group(var1 var2 var3) where var1 - var3 are variables for country, sector, year and then use this as my fixed effects.
So what I try to do is the following: xi: glm dep_var independent_var i.new_var, family(binomial) link(logit). Is this the right way to include fixed effects in a glm?

This method gives a long list of coefficients of the generated dummies in which I am not interested and I do not know how to eliminate them. absorb() is not option. I am really sceptical about how correct this approach really is. At this point I would like to state that my dataset is not a panel. I have observations in different years and countries for individual firms but each firm appears only once in the dataset.

Would vce(cluster new_var) be the closest that I can get to a fixed effects model?
How can the constant be interpreted?

Thank you,
E. Moon