Without a hands-on inspection, the best way to tell if it was originally a full-stocked rifle is to examine the barrel out beyond the rear pipe. If the gun once had a full-length stock, there should be a faint stock line running out along the barrel's two side flats. The lower part of the flat below the stock line should be cleaner/lighter/less rust, while the part above the stock line should be darker from oxidation, dirt, and general exposure to handling, etc., over the years. When enlarging your pictures, I don't see an iron under-rib on the bottom side of the barrel, which suggests the gun was full-stocked. If so, there should be a couple old scars on the bottom flat of the barrel from where the original barrel loops had been located.
The gun could be restored, but it would involve having the barrel "stretched" back to its original length by adding a piece of old barrel onto your breech end where it had been previously cut off. An experienced collector or restorer should be able to determine the original length of the barrel by removing the current barrel from the stock and lining up old barrel pin holes with the original barrel lug locations on the bottom of the barrel. The missing forestock wood is then replaced while retaining the original rear portion of the stock. But good restoration costs money, and you must consider the value of the gun you have, what restoration will cost, and the gun's potential value after restoration. In this case, with a relatively plain gun and no maker's signature remaining on the barrel, the value of the gun after restoration may not justify doing the needed work. While there is a range of prices for doing this type of work, I'd guess the barrel work would be about $400 for a good job, and the stock work perhaps another $300, if you use a good restorer, but not one of the top-level restorers.
To locate people doing restoration, you might try posting a couple good pictures of your rifle and describe the work you want done in the appropriate forum on this site, as well as the appropriate forum on Americanlongrifles site. Modern builders/restorers watch these sites regularly and you should get some responses. It also helps to ask their location, and work with someone within reasonable driving distance.
Shelby Gallien