My sister spells her name the same way.
Unless I misunderstand from your pictures, it appears you are missing the legs and the front panel. With rare exceptions, the likelihood of determining the precise look of a piano of that vintage is probably doubtful. More often than not a model number wasn't placed on the piano and countless combinations of furniture styles were available, including the wood type, legs and carvings on the front and sides. Ironically, even with a catalog you would probably only be able to determine your model or style by matching it with a picture in the catalog, except you can't because the pieces are missing - catch 22.
Other than the satisfaction of restoring the instrument to its exact original look, replacing the missing parts with something similar should have no real negative effect such as value. You might be able to find acceptable parts from technicians or refinishers. Personally I have kept such pieces lying around for way too many years "just in case". Old pianos get scrapped but sometimes it's too hard to throw away those pretty pieces.
Regarding the numbers, you should know there was more than one Shoninger Piano. The 1950's date is simply too late for your instrument. The number for that date would be for the Shoninger name used by National Piano Corp. in New York. I would guess your instrument is a B. Shoninger (established by Bernard Shoninger in 1850) in which case serial number 35844 puts its manufacture date in 1905, in line with its size, style and even the rosewood finish. Also the 3080 is probably a production code, a number stamped on various cabinet parts to keep them matched up during the manufacturing process.