1) Horn angles and variations thereof appear to be well studied in conformal geometry, see e.g. the papers by Ladue, Mary Elizabeth "Conformal geometry of horn angles of higher order", Amer. J. Math. 65, (1943), 455–476, and "Trihornometry: A New Chapter of Conformal Geometry" by Edward Kasner, Proceedings of the National Academy of Sciences of the United States of America,
Vol. 23, No. 6 (Jun. 15, 1937), pp. 337-341

2) You can find more about them in the paper "The recent theory of the horn angle" by E. Kasner, Scripta Math. 11, (1945). 263–267, 53.0X. It is from 1945, but it should give you some idea about the directions of study in that area. 

3) Horn angles can be measured within the model of superreal numbers, see the paper by J. Bair and V. Henry "Angles corniculaires et nombres superréels" (French) ("Horn angles and superreal numbers), Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 1, 77–86.