As preliminary, first consider what operationalism and falsifiability mean.
Operationalism associates the meaning of a concept with a way to measure the concept. This originates with P. W. Bridgman’s The Logic of Modern Physics, New York: Macmillan, 1927.
Hasok Chang quotes Bridgman’s text, page 10, where Bridgman uses as an example the concept of "length":
In principle the operations by which length is measured should be
uniquely specified. If we have more than one set of operations, we
have more than one concept, and strictly there should be a separate
name to correspond to each different set of operations.
Sven Ove Hansson quotes Karl Popper’s idea of falsifiability in Conjectures and refutations. The growth of scientific knowledge, New York: Basic Books, 1962, page 39, as:
...statements or systems of statements, in order to be ranked as
scientific, must be capable of conflicting with possible, or
Those “conceivable observations” could falsify the “statements or systems of statements” if the observations actually occurred.
Given the above, here are some answers to the questions.
(1) Does operationalization guarantee testing falsifiability?
Defining the concepts one uses in terms of how they are measured does not mean that the statements or systems of statements using those concepts are falsifiable. In addition, one needs to describe conceivable observations that could falsify those statements or systems of statements using those concepts.
(2) Is the study of crowd behavior falsifiable science? From a
Freudian perspective I would say it is not falsifiable (e.g.,
collective unconscious) and from other perspectives that
operationalize some crowd variable I would say it is falsifiable.
The study of crowd behavior needs to construct statements or systems of statements sometimes called "theories". Those theories may or may not be falsifiable. If one can come up with a conceivable observation that would falsify the theory then the theory is falsifiable.
(3) Could ethical concerns convert a question from being falsifiable
into being non-falsifiable? Once we operationalize our variable, there
is another "obstacle". Ethical concerns do not allow us to conduct a
study that could arise for example panic. So sources to test our
hypothesis are limited.
It is not necessary to be able with current technology, resources or other limitations such as ethical concerns to actually observe what one could conceivably observe for the statement or system of statements to be falsifiable.
(4) Could an evaluated non-falsifiable problem evolve with time and
become falsifiable? (e.g., Freudian vision to cognitive vision)
Once a conceivable observation is constructed that would falsify a statement or system of statements then those statements or system of statements are falsifiable based on Popper’s 1962 description of falsifiability.
References used in this answer:
Hansson, Sven Ove, "Science and Pseudo-Science", The Stanford Encyclopedia of Philosophy (Summer 2017 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/sum2017/entries/pseudo-science/
Chang, Hasok, "Operationalism", The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/fall2009/entries/operationalism/