Good Evening, Steel Horse Bailey;
In answer to question #1:
The US Army Uniform Regulations of 1861, Paragraph 49 state:
"For fatigue purposes, forage caps, of pattern in the Quartermaster General's Office: Dark blue cloth, with a welt of the same around the crown, and yellow metal letterns in front to designate companies."
This may have worked on parade, and for inspections, but wartime photographs seem to indicate that the bright brass over the forehead was removed fairly quickly on the battlefield.
From the US Army Uniform Regulations of 1872:
"For all Enlisted Men: Of plain blue cloth, same pattern as for officers, with badge of corps or letter of company of yellow metal worn in front as for officers."
...nothing to indicate if both were worn at the same time, but this was likely left up to the post commander.
From the same regulations:
Fatigue hats:
"For Officers and Enlisted Men: Of black felt, according to pattern, to be worn only on fatigue duty and on marches or campaigns."
...which answers your second question: Cowboy-type hats were first authorized in the regulations of 1872.
You might want to pick up a copy of John P. Langellier's Army Blue: The Uniform of Uncle Sam's Regulars, 1848-1873; this volume, and its companion (More Army Blue, by the same author) is chockablock full of period photos, and photos of existing relics, as well as copies of the various uniform regulations.
I have the honor to remain,
Your Obedient Servant,
Bvt. Capt. Malachi Thorne