The answer to the second question is **yes**, for any non-flat triangle $T$, the set of angles $A$ is dense in $(0, \pi)$. This follows from a stronger result of Barany et al. [Theorem 1, 1]:

>> **Theorem**. Successive barycentric subdivisions of a non-flat triangle contain
triangles which, to within a similarity, approximate arbitrarily closely any
given triangle.

Here is a nice divulgative [article][2]  on the dynamics of iterated barycentric subdivisions. It also has a list of further readings on the topic. 
(*The joy of barycentric subdivision*, by Bill Casselman)- 
 
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[1] I. Barany, A. Beardon and T. Carne, "Barycentric subdivision of triangles and semigroups of Möbius maps", 1996. [MR1401715](http://www.ams.org/mathscinet-getitem?mr=MR1401715).

[2]:http://www.ams.org/publicoutreach/feature-column/fc-2017-06