Howdy
Without having something to scale them against, it can be tough to tell from a photo if it is a 32 or a 38.
The 38 Double Action at the top of this photo is a 4th Model, it shipped in 1898. The 32 Double Action at the bottom of the photo is also a 4th Model, I have not looked up exactly when it shipped.
Judging from the size of your cylinder in comparison to the frame, I suspect you have a 38, not a 32. Serial Number range for the 3rd Model 38 is 119001 through 322700, manufactured from 1884 -1895. The only problem is your front sight should be pinned to the barrel like the two revolvers in the photo above. Your front sight has been forged in place as part of the barrel. The 5th Model had either a pinned front sight or a forged one. Perhaps somebody swapped out your barrel for a 5th Model at some point. Who knows?
The cartridge it is chambered for is 38 S&W. Not 38 Special.
If a 32 it is chambered for 32 S&W. Not 32 S&W Long, that cartridge had not been invented yet. Certainly not 32-20, the cylinder is way too short for that.
Yes, S&W was nickel plating revolvers from the very beginning in 1857. In fact with their small revolvers more left the factory nickel plated than blue. 19th Century blue finishes were not as robust as modern blues are. The old blue could rub off fairly easily, that is why nickel plating was popular, it provided better corrosion resistance than some of the old blue finishes. Case Hardened hammer and trigger are correct for Nickel Plated S&W revolvers, notice the colors on the triggers and hammers in the photo above. Trigger guard is blued, which is also correct. Blued latch near the hammer is also correct for a Nickel Plated S&W.
The grips are Hard Rubber. Be careful, they get very brittle with age.
I suspect your revolver may have been refinished at some point. The joint around the side plate should be almost invisible. Still, it is a pretty good job. Often the edges of the side plate get rounded over when polished for a new layer of nickel, and then the joint stands out like a sore thumb. I can't tell from your photo, but if the marking on top of the barrel rib looks a little bit 'washed out' in places, then almost certainly a refinish. If refinished, it is a pretty good job.
The nickel plating on this 38 Double Action 3rd Model, which shipped in 1888, is a bit more challenged than yours. Yes, the screws should all be nickel plated, but it is not unusual for screws to be replaced over the years, with whatever happened to be handy when needed.
As far as ammunition is concerned, none of my 19th Century revolvers ever get fired with modern Smokeless ammo. Only ammo that I load with Black Powder or a black powder substitute.