Nobody can say "they always did it this way" or "they never did that" when it comes to actual Hawken rifles. Many of these rifles were built to the customers specification.
But you will find traits that are particular to the Hawken/Gemmer shop.
There are examples out there that show various treatments.
But the above post is on the right track, it seemed case hardened furniture including the breech and tang were common. This was not always color case hardening, remember these were utilitarian rifles and not many "fancy rifles" were likely built. Much of the case hardening was done just for the hardening factor and the parts showed a grey color. Wedge escutcheon plates and screws were likely left in the white, they were finish profiled as a part of the stock and unlikely removed for final finishing.
Fasteners (buttplate screws, tang & lock bolts etc) were likely heat blued, no call to spend the time rust blueing them.
Barrels were likely rust blued, whether or not browned barrels were produced is not a certainty, they may have been browned or the brown is a progression of the blueing and exposure to the elements. Remember, blue and brown are a mechanism of the rusting process, blue is achieved by "killing" the rusting process early.
I believe there are a couple of original examples that show evidence of the barrels having been in "the white" i.e. unfinished.
W Selb, D Taylor Sapergia build contemporary rifles that are very true to originals, take a look at their work.
Don Stith is an encyclopedia on the rifle as well.