Going by the pictures, and the info you've provided, my SWAG is that you may have a No. 40 New Model pocket rifle with a cut back barrel. 

Also appears that the rear dovetail sight, which should have been a tangent style, was swapped out with a Stevens #106 leaf rear, or a Lyman #6 rear that is missing the "bar" leaf. 

I'm leaning toward the Lyman #6 with the missing leaf, as the single leaf appears to be slightly offset forward of the pivot screws, though my photogrammetry skills may be a bit out of alignment today. 

And this SWAG is worth exactly what you paid for it.   

Keep the pictures coming. Definately interested. I have a similar gun that I am currently working on. I think mine was also shortened to 8" as I recall. John Taylor currently has it making a new extractor. I had him reline mine and recomend him highly if you choose to reline it. Stocks are hard to find but are out there. I bought the stock for mine on GB because the price was right and ended up buying the gun it came off from the same guy and ended up bringing it back to a shooter. These are neat period pieces and fun to shoot. I have a really nice one in .32 Rimfire. Ammo cost is a problem with the .32 but fortunately I bought some Navy Arms ammo back when it was available.
Keep us posted and have fun with it.

Slumlord, there's a  Pocket Rifle in 32 in my neck of the woods about 95% condition with the stock and a velvet lined custom case--asking price is @ $1100.00--but was a little steep for my gun budget right now (bought a few too many High Std. pistols recently--one of my other weaknesses)  and, like you, I like to shoot my guns and didn't want to get in to having to source 32 ammo or converting it to 22.
Come to think of it, I know of 2 Favorites in 32 as well at KTP in Me., maybe someday someone will make another batch of the stuff for guys like us that like to shoot them.

PS--have John check the firing pin bushing too.  Took it out for the first time tonight and the "lip" on the bottom front was broken off--that forms the slot the screw goes through to keep the bushing from just coming out--yet it had functioned OK for about 150 rounds so far.
It was too small to silver solder back on, so I just built up some phosphor bronze / silver alloy solder and filed it back to shape--so far, so good!

Work in progress--here's some more pics;
I have the barrel stripped and polished, and playing with the sights;
First, the pics actually make the parts show up a little worse than they look in real life--some of what appear to be pits and file scratches remaining is really just dust and fingerprints--I was in a rush to do the pictures.  Notes are;
Barrel was polished starting with 220 sand paper, then 400, 600, 1500 wet or dry paper, then Simichrome polish on a rag, by hand.  I use my favorite tool--a piece of paint stirrer stick with the paper wrapped around it to help apply more pressure more evenly.  I seems in one place already filed through the nickel long ago, that if I polish it well enough the polished steel blends right in with the old nickel.  Of course at some point I may well send it out to be re-plated anyway.
Barrel--waiting for some Brownell's Oxypho to come in, so hopefully by next Wed. or so the barrel will be blued.  I trued the muzzle with a file (it was 1/16" off with file marks under the patina from an old cut back job) and I crowned it with a round head screw in a drill using Napa water based valve grinding compound, then polished with 400 grit wet or dry wrapped around a sharpened pencil.
Sights--front sight is an unknown 3/8" dovetail metal sight I had in kicking around that looked to be a good height--used a file to trim it down to the 5/16" dovetail on this gun.  Sight is not final fitted yet--any misaligned grooves you see will be taken care of before final polish, blue and fitment are done. I will round the corners to blend with the barrel.
Rear sight;  The original rear sight leaf was loose and floppy and screws were stuck.  I heated the sight gently with a torch three times a day for 3 days and then soaked it in a mix of 1/3 Liquid Wrench and 2/3 Kroil after each time, then let soak for 2 more days--the screws came right out.  I may go to Marbles adjustable sight as a shooter for better sight picture. Pics to follow in the next post

Steve,
Nice pistol and nice clean-up without being over agressive. Did you notice any lateral loosness of the barrel at the pivot point? Question based on experience with a similar type that was difficult to keep tight with use.
Chuck

Shooting single shot pistols.  Right handed shooters. Place the pistol across the left forearm grab the your shirt push forward with the left arm and pull back with the right.  You will have a very solid off hand position.

Det

Steve,
Thanks, says something about the quality of the Stevens. The one I was messing with was similar but not Stevens.
Chuck

More Pics;
Also, when I detail stripped the guts and got all the deposits out of the action, the trigger pull went from a nice clean break at 15 oz to a gritty creep at 1 lb 5 oz.  It seems some rust on the hammer slot for the sear was gunked up and rusted so the sear had very little purchase on it--go figure!.  Went back in to the sear and hammer today and got the trigger to a smooth pull at 1 lb. 2 oz with just a little creep--I'll leave it like that and well enough alone, at least for a lttle while.

Two steps back I think--the lollipop sight and the trigger pull and before that the broken firing pin bushing--had to fix it again today with silver solder (for a better weld), but no biggie, I figure I can always make a new one from scratch out of a drill shank.
But, all of that I have fixed or can fix but the lollipop is going to be just plain tedious I am sure.  It's a PITA to get in to a gun to make it better and work again and  then find hidden things that need work.--kind of like working on old cars or old house plumbing.  But, I like to shoot them, so better fix it now than later on.
Got the barrel blued today and it's curing in oil for the night. Will assemble tomorrow, take to the range to fit and file the front patridge sight and shoot for group, then post some pics tomorrow night.  Guys at bullseye league will get a kick out it on Thurs. night I am sure!
Overall goal is to have a nice shooter that looks like a well cared for 80 or 90 year old gun that that has been shot and has a few character marks, not one that looks unfired and was sitting in a safe all that time, plus have fun with it and make it a nice compliment to my 418 that is about the same original condition as this gun will be when I get done with it.

Pic's
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Hmmm--thanks for the input on the firing pin and "bushing".  The reason i thought they were two separate parts is the pin part that projects through the breach face is the small pin on a very thin base--picture the pin popping up from the center of a small sliver of a disk the same diameter as the base,  that is only maybe 20 thousandths thick. Definitly two parts--I don't see how this little washer or disk type piece with the firing pin projecting from the middle of it could have separated from the end of this base.  Maybe it's already been worked on and fixed with this separate piece?  Wondering now if I should silver solder or soft solder this piece to the base in the correct positon, or just leave as is.  I don't have a lathe though if I wanted to spend the bucks i could have my smith do it.
Maybe try to fix myself and could always have my smith make one if need be.  I might try using a small drill shank the diameter of the firing pin, and drilling a hole in the face of the base and soldering it in  and trimming to size, for lack of a lathe. 
Here's a thought--if your 40 and Sendaro's both had firing pin breakage and maybe mine too, maybe that was a common problem on these guns, and some old time smith "fixed" mine by making this "floating firing pin" that floats on the face of what I've been calling the bushing (as on the picture)
Thoughts anyone?

Thanks for your info--I hope the moderators dont move this to the gunsmithing section! (If the do, I'll pick it up there)
 
Interesting--I thought it was made in two pieces because I assumed if the firing pin tip had broken and dryfiring continued, I'd have a dimple in the breach face of the barrel from the firing pin hitting it, much as a too long firing pin will do on many rimfire guns--as there'd be nothing to keep the pin from travelling too far.  Yet there is no dimple at 12 o'clock on the barrel breach.  Yet, I know this gun was given to a 5 year old boy by a note I found tucked away in the grip.
I think I'll try regular or silver solder on it first, then try drilling it and inserting a small drill bit shank in the hole and triming to fit if the solder job doesn't hold.  Kind of like doctors motto  "Do the least invasive procedure first".
I'll take a picture of the original pin and base in two parts so everyone can see what I'm dealing with
Thanks again !!

Steve, you have an excellent point! There was no dimpling of the rim area of the barrel breech of my gun, either. And I now realise that it could not have occurred, for the firing pin does not quite reach that surface in its forward-most position. Therefore, the breakage and peening of the tip base must have occurred during live firing.

As Chuck says, the firing pin is very easy to make. It should be done on a lathe, as I did; but a craftsman who is good with a file could probably do it by chucking the work piece in an electric drill that is securely fastened to the work bench.

418Steve, FWIW here is what I have experienced with the 40s. I have only 2 in my collection, but a close friend of mine has 20 or more. I've looked over the examples in his collection and did not note any with broken firing pins. Some did have burrs from dry firing though.                                                                                         One of mine had a broken firing pin in it when I bought it. Turned one and fitted it to the standing breach. Made it from tool steel but did not harden it. It is working just fine. However there was a peened burr on the edge of the chamber from dry firing. I used the simple tool that Brownell's offers to iron out the burr. It worked like it was said to and all went well. Soldering your old one back together is not going to work. Have a new one made and take the time to fit the notch to the retention screw in the frame. You want that notch just wide enough that it will allow the firing pin travel to the rear to full retract the tip of the pin and forward enough that it make proper contact with the cartridge base without the notch slamming against the retention screw. Hope this helps.

Are you still in need of a butt stock? Please advise, as I may know where one might be purchased.

Update on the gun and for those who may have a broken firing pin.

Finally made a new front sight post to replace the one that had rusted off the lollipop.  Got a 4X48 screw, mounted it in a dremel and put the dremel in a vice.  Then held a regular then a small diamond file against the screw to remove the threads and contour the lollipop tip and then polish with 600 grip paper. Then cut to length and used a .020 thick dremel diamond cutoff tool to put the screw slot in the shaft.  I lucked out and got it right the first time!

Second, made a new firing pin based on what others told me it should be like.  Old one was broken into 3 pieces as shown.  Turned a 1/4" drill shank down to size, about .225", then drilled a hole in the face and inserted a 5/64" drill shank, silver soldered in and cut to length.  The original firing pin was 1/16" dia. so this is just slightly oversize--but it needed to be as the firing pin hole was worn and this took up the slack for a nice fit.  Also, now the firing pin hits to the very edge of the cartridge head, whereas the old one was hitting slightly away from the very rim due to worn firing pin hole and the original 1/16" diameter.  Have now run 150 rounds of CCI Std. Vel ammo through it with not a single misfire.  This took me two tries to get right. Problem solved!
Next order of business is still to get a repro or OEM wire stock and rear sight (thanks to some leads Sendaro).  Still pretty sure I'll leave the barrel as is--wont make a decision on relining it until i get the other parts first. Yes, the firing pin is offset a bit to the top to better align it in the worn firing pin hole.
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Nice work. Amazing what you can do with the right tools and a lot of patience.