Questionnaire Analysis virtually without continuous Variables

Hello!
I am doing an analysis on a questionnaire of hunters taken in 4 different
districts of some mysterious foreign country. The aim of the study was to
gather info on the factors that determine the hunting success of a
peculiarly beautiful bird in that area. All variables are factors, i.e. they
are variables such as "Use of Guns - yes / no", "Use of Dogs - yes / no" and
the likes. The response is upposed to be "number of Birds caught", which was
designed to be the only continuous variable. However, in reality the number
of caught birds is between 0 and 1, sometimes hunters answered with 2.
Unfortunately, it is not the questioner who is burdened with the analysis,
but me. I am struggling to find an appropriate approach to the analysis. I
don't really consider this as count data, since it would be very vulnerable
to overinflation (and a steep decline for counts above 0). I can't really
suggest binomial models either, since the lack of explanatory, continuous
data renders such an approach quite vague. I also struggle with the random
design of the survey (households nested within villages nested within
districts). Adding to that, hunters don't even target the bird as their
prime objective. The bird is essentially a by-catch, most often used for
instant consumption on the hunting trip. I therefore doubt that any analysis
makes more than a little sense, but I will not yet succumb to failure. Any
ideas?

Thanks in advance!

PS: I just realized that this is not a question related to R but to
statistics in general. Apologies for that!