Dear Roberto,

I would say the answer is definitely 'yes', but you will need to discuss the mixed sampling approach when you interpret your analysis. Obviously you will need to add a grouping variable to indicate which group each individual belong to.

Yes, but how to do this is the question.

If the people in the group of non protester (NP) were selected via a probabilistic sample, then I suppose that you put a question that filter out any protester of the selected sample and that you have corrected the sampling weights because of them. So you would have a national sample of NP.

If the systematic sample of the protesters (P) was done with a random begining, it could be a probabilistic sample, but in this case the population would be of those who were protesting in this particular ocasion. Then there is a third subpopulation of people who has done some protest in the past but that they were not protesting for this particular ocasión, and from them you do not have information (PP: Protester non Present).

You could expand your first sample, using the weights derived from the sampling procedure of the NP subpolulation, and produce estimators of each asnwer proportion. You may estimate the same proportions for the P subpopulation without weighting, and you may perform test of hypotesis about the differences between this parameters from the population NP versus P. But you can not test for differences between them and the subpopulation of PP because for this population you have not information. To obtain information about PP you may use a sample similar to that of NP filtering in only that people who some time did protest but were not present in the protest from wich you take your systematic sample.

I would say yes. The problem is that the "difference" between the two groups (protestor vs non-protestor) might be caused by different biases in the sample. So you surveyed protestors at the animal rights protest and your non-protestors were all breeding roosters for cockfights. Or maybe many of the non-protestors were all back from protesting tax code changes at the city council meeting.

That said, I don't think you had much choice. The solution is not with the sampling strategy, rather it is asking enough questions that you can address some of these kinds of issues (you won't get all of them, but at least you considered the problem). Are there factors that tie these groups together that are unrelated to the core issue you are trying to study? I would think that the protestor group is at greater risk of having hidden factors in common. Thinks like the unemployment rate in the protestor group was 30% greater than the population at large, or the protestor group had a median age of 55 while the population at large had a median age of 38.

An example is stratified sampling from two strata. Yes, it is possible.

Sometimes No.  Deming and Gleser (JASA 1959) solved a problem that had been open for more than thirty years.  The issue was using samples for certain issues related to sampling overlap.  Although they gave a full theoretical development, generally the variances of estimates were much too high.  Their cautions should extend to may other situations when sampling is used.