I believe I have a gun dating between 1884-1885. On the barrel it says “L. C. Smith Maker ; Syracuse N.Y.” Serial number is 13176 E. I would like to find out more information about it as well as if there is a source that I can locate a missing hammer for the left side.

Michael there are no records of when these guns were made but in 1884 the serial numbers started at 10000. Also it is hard to speculate if the serial numbers were sequential, but I believe that your gun was made either in late 1884 or early 1885.
The lowest grade was the Quality F and the Quality E had some engraving, finer checkering and had Fine Damascus barrels. The highest grade was the Quality AA and there are only a couple known and one of which I have the serial number to.
Finding a hammer is going to be tough but I would keep looking at eBay for one as they do show up there at times. That style hammer would be the Type 1 and was used until 1896 around serial number 545xx,

If you would be so kind as to mention the gauge and barrel length I would appreciate it for my records, thanks.

Michael, there are no records remaining for these guns. Your hammer gun was probably made 1885-86. The "E" was the second lowest grade at the time. Our friend, David Williamson, tracks all of these old hammer guns so if you could post barrel lengths and type of damascus barrel and gauge, it would be helpful to him. David may chime into this post with more info. I have several of these in 10 and 12 gauge and find them to be very well built. I sjoot them on a regular basis. Enjoy your find

For barrel diameter determine gauge, should I measure where the shell is inserted or at the end of the barrel where the shot exits?

Doing it at the muzzle end could be easier. For a 12 gauge it could be from .690-721 or so. For a 10 gauge it could be from .720-.750 or so.
An old rule of thumb was if a dime fit in it was a 12 gauge.

My gun is a 10 gauge and the barrels measure 30 inches in length.

The muzzle end measures .760-.765 according to my calipers.

Michael thanks for the information.

Michael
If you are still looking for an early style left hammer I have an extra that would work for you.
My email is runninforthebarn@gmail.com

Michael, just to clarify, are you measuring the inside diameter or the outside diameter at the muzzle?