> The median is specifically the mid point of the data set, so it won't be "any number between 0 and 2", it will be one specific number depending on the data set.
No. The median is not necessarily unique (just like the mode, and unlike the mean). For more information, you can check here: https://en.wikipedia.org/wiki/Central_tendency#Uniqueness
A median (and not necessarily the median) is any point that minimizes its distance from a set of points under the L1 norm and often you have several points that minimize such distance (such as in the example you provided, assuming perfect 50/50 split between men/women).
> If you look at ten people with 2, ten people with 0, and one person with 1, then the median will be 1.
Correct.
> If you make it twelve people with 2, still ten with 0 and one with 1, suddenly the median is 2.
Sure.
But, again... you are choosing cases where the median of a set of points is unique (odd number of points aligned along a single dimension)... this is the exception, rather than the rule (particularly in higher dimensions than one).
> Across an entire population, I would hazard a guess the median would be 0 or 2 unless the difference in numbers of each sex is small enough that people with 0<X<2 testicles might fall right in the middle of the distribution.
Yes... as I said. The actual median is not 1, but actually either 0 or 2 (because the ratio between men and women is not exactly 50/50). But, again, even in the (rather unlikely) case that the ratio between men and women would be exactly 50/50, the median would still not be uniquely defined as 1 (in such a pathological case, any number between 0 and 2 is a median).
EDIT: after re-reading your comment carefully... I have not considered the subpopulation of people with a single testicle (which is nonzero); so I stand corrected... the median could be 1 (but it is very unlikely that it is the case... it's probably either 0 or 2).