One statement and six questions here.
But first on behalf of namesakes everywhere and the continuing history of statistics, please note that the proper-cased name "Box-Cox" is standard.
The Box-Cox transformation transforms our data into a normal
At most, that's the goal. It can't always be achieved, even approximately. For example, a distribution that is in essence a series of spikes can't be transformed into anything but another series of spikes.
How is that even a proper technique?
Conversely, in what sense is it improper? The general idea of transformation is that it can be easier to see and analyze what is happening on a transformed scale, while specifically there are many techniques for which some approximation to normal distribution(s) provides, if not conditions that are assumed to be true, as so often stated, then at least relatively ideal conditions for summary and inference. Note that generalized linear models borrow the idea of fitting on a transformed scale without actually obliging transformation of the response variable.
What if our data didn't come from a normal distribution?
It's not clear what the puzzle is here. It's precisely when data are not normally distributed that the question of whether there is a simple transformation to normality arises.
How could someone just blindly apply the Box-Cox transformation?
As above. Some people blindly apply every statistical technique they use and statistical people tend to disapprove of that rather than approve. At the same time, life is short and there is an element of trust in most technique use, as no-one can derive and justify everything they do.
The other questions look like the same questions rephrased, or else I am missing nuances. But in turn I'll repeat what seems to me a simple key: normal distributions are often an ideal, yet many techniques work well even if that ideal is not satisfied.
At this distance, the main contributions of the Box-Cox formulation from 1964 appear to me to be
The idea that the data themselves will tell you which transformation is most nearly appropriate. (We should add that sometimes no transformation will help enough to be worth applying.) Box and Cox formalised that data-guided choice of transformation in various ways, but the important point is implicitly or explicitly to try out various transformations systematically. (All too often, search for transformation appears to be stabbing in the dark, as when people tell you that they have tried logarithms and squaring, but nothing works.)
The idea that most of the transformations in use, especially for positive measured variables or counted variables, belong to a family including not only the powers but also logarithms. This idea was also widely emphasised earlier, notably by Tukey (1957), whose paper was rather oddly not cited by Box and Cox, but the Box and Cox formulation, followed by Tukey's later work, seems to have been more successful in popularising the idea of a family. As just stated, emphasis on choice from a family makes the idea of transformation choice more systematic, and less ad hoc. Note that Box-Cox is indicative, not commanding, on what the decision should be. In their own worked examples they choose logarithms and reciprocal transformations, thus rounding off the powers given by their estimation procedure. Indeed both examples were of the kind where experienced analysts would have chosen the same transformation any way before their paper.
Box, G.E.P. and Cox, D.R. 1964. An analysis of transformations. Journal of the Royal Statistical Society Series B 26: 211–252.
Tukey, J.W. 1957. On the comparative anatomy of transformations. Annals of Mathematical Statistics 28, 602-632. doi:10.1214/aoms/1177706875. http://projecteuclid.org/euclid.aoms/1177706875.