You have what is called a 32 Double Action, Second Model, shipped around 1891-1892. It fires 32 S&W caliber ammunition. Correction As Tom pointed out, I missed the fact that it has 6 holes in the cylinder, so it is a 44 Double Action revolver. The cylinder looks short and if it is 1 7/16" long, it is a 44 Russian caliber. If it is a 44 Russian, it would have shipped around 1881. The other caliber could be 44-40, but that would take a 1 9/16" cylinder. If it is a 44-40, it would have shipped around 1888. Check the cylinder length and that should give you the caliber answer.
There is quite a bit of wear on the gun and chances are that the internals are either very dirty or worn. I would first remove the sideplate and either soak the revolver in a solvent to loosen up the crud, or use an aresol carberator cleaner to spray the mechanism. After it is cleaned and dried, lubricate and try the action again. Hopefully, that will solve the single action operation and yes the revolver should function in both SA and DA. I assume it works in double action?
Forgot to mention that you should remove the hard rubber stocks before cleaning. Take care in removing and do not pry them, since there is a stud pin in the bottom of both stocks holding them in place.