1861 Springfield replica or original?
We always used a simple no.2 graphite pencil for small rust areas. This will remove light surface rust, but won't inhibit rust. Any pitting there cannot be fixed satisfactorily, and attempting to do so may end up looking worse than the pitting. For re-enacting purposes, go over entire firearm with clear Kiwi shoe polish to protect from weather and rain. Works pretty well. Water just beads up similar to how car wax looks after a rain.
About staining, don't know about that. If the rifle is original, don't mess with it at all except for three-in-one oil, which should be ok. That would be a wall hanger anyway.
Les