Your mantel clock was manufactured by The E. Ingraham Clock Company, Bristol, CT. You can read more about this company here.
Looking at the dial of your clock suggests the manufacture date in the late 1800's up to about 1910.
The small thumb wheel under the "12" is to adjust the clock to run faster (F) or slower (S). As far as the gong, you are correct, the bell dings on the half-hour, and the gong counts the hours. The fact that yours is not correct is not a big problem. These old clocks used a count wheel and can easily get out of sync for many reasons.
There are several ways to put the clock back in sync, but the easiest way to explain here is to wait for the clock to gong on the hour (say 8:00). Then remove the minute hand and re-install it pointing to the 12. Then carefully move the hour hand to the correct hour (the hour hand is held to the shaft by friction and can be moved to the correct hour). If moving the hour hand causes it to become too loose, just push it toward the dial to tighten it. In some cases, you may have to remove the hour hand and squeeze the collar to make it tight again. It sounds harder than it is.