Somewhere between 1948 and 1951 according to the S&W catalog #4.
That is correct. According to the Standard Catalog of Smith and Wesson, the serial numbers for 1948 through 1951 were C1 through C223998. I would say with such a low SN yours probably left the factory in 1948.
The proper name for your revolver is 38 Military and Police (Postwar). In 1957 S&W changed over to a model number system, and the 38 M&P became the Model 10. But many shops and owners often mislabel these revolvers as Model 10s.
The numbers on the frame under the yoke are assembly numbers, nothing more.
The great majority of S&W revolvers do have the serial number on the butt, but not all.
The I frame 22-32 Heavy Frame Target revolvers known as the Bekeart Models had an extended grip that covered the bottom of the grip frame.
Like this:
So the serial number was stamped on the front surface of the grip.
This was also true of the I frame 32 and 38 Regulation Police revolvers. This is a 38 Regulation Police.
Beginning in 1957 S&W began stamping model numbers on the frame under the cylinder yoke. This is a Model 10 dash 5. The dash numbers represent a progression of engineering changes.
With everything I've been learning about this gun I kind of forgot to ask -- it's still ok to shoot even though it's probably 72 years old, right?
Probably. If you are not familiar with these revolvers I would have a
competent gunsmith who is familiar with them check it out first. I have lots of old Smiths, and I shoot most of them, but I know what to look for. Stick with standard velocity and pressure 38 Specials. I would not shoot 38 Special +P in one that old.
By the way, the Ampersands (the & symbol) on the barrel of your 38 M&P are what collectors call the 'lazy ampersand' because of the way it sprawls out and looks like it is laying down.
This Model 14-3 from 1974 has the more modern Ampersand.
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