Buglers Jacket, Federal Horse Artillery
credit: Time-Life: The Civil War Series
bonus: He is buried (along with his brother) at Oconee Hill Cemetery, a short walk from the 100,000 seat University of Georgia Bulldog football stadium in Athens, GA.
Edit - Although Cheatham Hill assures us that the jacket pictured in the question was, indeed, a bugler's jacket, the picture in post # 4 (and at least one other post) shows a guy dressed in an identical jacket and holding a snare drum, so evidently that style of jacket was not limited to use by buglers exclusively. I will accept either "musician" or "private" for the rank.
Technically, in today's U. S. military, the "branches of service" are the Army, Navy, Marines, Coast Guard, and Air Force. During the Civil War, you would have had the Army, Navy, and Marines, and a case could be made for the Revenue Cutter Service as the forerunner of the Coast Guard and the Balloon Corps as the forerunner of the Air Force. So I will accept "Army" as a correct answer for the branch of service, although clearly the desired answer was "artillery," which is a branch of the Army.
The bonus question asked for an "edifice" located a "short" walk from Cobb's grave. The definition of what distance should be considered a "short" walk is open to a wide variety of interpretations, so I will be liberal in accepting various buildings around Athens, GA as correct answers. However, according to my interpretation, an "edifice" is a single building, so I am not going to accept any answer that includes multiple buildings (such as "the University of Georgia"), nor am I going to accept the name of the cemetery or any monument in the cemetery as correct answers.
Hoosier