Etymology of Dogie has appeared somewhere in a thread, but it would take a search to find it.

Dogie 1 (preferred). Derived from Spanish dogal, a motherless calf. See Lingo and Definitions, Chapter 9, "Cowboys-Vaqueros, Origins of the First American Cowboys," Donald Gilbert y Chavez. Cowboys Vaqueros

This excellent booklet, on the University of New Mexico website, is worth copying for reference if you are interested at all in cowboys and their history.

Dogie 2 (preferred by those with Texas traditions). From some cowman, orphaned calves received the name 'dough-guts,' signifying a lean, pot-bellied, malnourished calf. Unbranded calves, without brands, were claimed by the first to find them. The term was later shortened to 'dogie.'
This explanation in Ramon F. Adams, "Western Words," p. 51-52.

Dogie first appeared in English print in 1888, in Cent. Magazine, as 'Texas 'doughies,'" but anecdote has it that the term is much older. The Spanish 'dogal' is centuries older, with the original meaning of halter.