It looks like a Hawken to me, judging by the brass nosepiece and thimbles, the shape of the brass trigger guard, and the single barrel wedge. It's definitely not a Mountain Rifle, I have a MR and posted photos in my intro, so you can compare.
For the rusty barrel, I would try cleaning it with Evapo-Rust, just like you would do with a regular cleaning, then rinse and dry it and see what it looks like. I suspect that the Evapo-Rust contains EDTA or something similar, which will dissolve existing rust but will not harm the underlying steel, and it is non-corrosive. This product should work in just the length of time that it takes to do a thorough cleaning by pumping the liquid up and down in the bore. Then you can use a penetrating oil of your choice leaving it to soak, then see about fixing the drum.
I would give the barrel an acid browning treatment with the product sold by Track of the Wolf instead of blueing it. That brown color treatment will cover the surface rust trace pitting better than blue. Look at the photos of my rifle to see what the acid brown color looks like.