Thanks for posting and welcome.
#1 - 2nd Model SMALL FRAME HAMMERLESS 5th Variation (barrel rib stamping with state not spelled out, i.e. MASS.) made to handle smokeless/Nitro powder (cartridge stamp on left side of barrel indicates this. That barrel rib stamping was only used in 1915 - so, guess what year your gun was made.
Standard issue is: 3" barrel; nickel plated; 5 x .32 S&W (the short one) or 7 x .22 short or long (will also chamber the now obsolete .22 S&W Long RF) but not the .22 long rifle; molded, black hard rubber grips panels with "target logo." Options: barrel lengths of 2" (BICYCLE Model), 4", 5" or 6"; blued finish; checkered walnut, Mother of Pearl or (RARE) Ivory.
With all those matching serial numbers and parts of the serial number - your gun is all original. The 'H' stamping under the top strap is ONLY found on the small frame HAMMERLESS guns. My conjecture - these upper barrel assemblies were made and stockpiled for production runs and the 'H' was stamped to differentiate those intended for use on the HAMMERLESS from the same size uppers used to make the hammered PREMIER model.
Current RETAIL values: 100%/NIB - $265; 95% - $200; 90% - $160; 80% - $145. All values are +/- $20 depending upon area.
#2 - This is a 2nd Model YOUNG AMERICA DOUBLE ACTION "small frame" made to handle the .22 rimfire short or long black powder or semi-smokeless standard speed cartridges. It will also chamber the now obsolete .22 S&W Long RF but not the .22 long rifle. This model was in production from 1905 to c. 1941 and along with it's sister "medium frame" YADA chambered for the .32 S&W close to 1.5 million were produced. With your s/n in the 335,000 range a quick extrapolation seems to indicate your gun was made sometime between 1910 and 1914. The are currently no production to s/n records.
Standard issue of the small frame: barrel length 2" (approx.), Nickel plate, molded black hard rubber grip panels with "filagree"/floral pattern. Options are: barrel lengths of 4 1/2" or 6"; blued finish; grip panels of checkered walnut, M.O.P. or Ivory (very rare).
Current RETAIL values: 100% - $235; 95% - $180; 90% - $150; 80% - $125.
As with the SF Hammerless - expect some value or price fluctuations depending upon area.
These little guns seemed to have been very popular as vest pocket guns and in the case of the YADA as boys' noise makers for Holiday celebration - in fact period advertising makes a big point of this.
#3 - hrf nailed that one.