So, if it was "absorbed" by Universal Match, in the mid 1930's, does that mean that this book pre-dates the absorbtion?

According to lvstriphistory.com The Red Rooster originally opened in 1931. Our earliest phone book is 1939, and doesn't mention the Rooster. The Red Rooster IS listed in 1944, with a more 'normal' phone number of 1177-M.