I can't figure out if this is a real 1804 Large Cent. I have taken it to a few coin shops and they feel that it is an 1804. The date is kind of hard to see in the picture. Also, I don't think it is a "restrike." But I don't know. Here are the pictures. They arn't the best in the world but hopefully they will do.
It has all the hallmarks of a once circulated large cent...if a dealer at a coin shop reports flat out it's a not a re-strike or counterfeit, it's probably a mint made product. In hand, you can tell much faster than looking at some images. It looks real enough for me.
It is severely damaged, it has red and green staining, tons of damage, shown by pits all over the coin. Believe it or not, I have the same coin, but in worse condition
It is NOT an 1804 large cent. The 1804 comes from just one pair of dies. Two things I saw that immediately told me it wasn't an 1804, on the 04 the gap between the ends of the wreath is just a little right of being below the center of the E. On your coin it is a little left of center of the S. On the 04 the point of the outer leaf on the second leaf cluster is below the edge of the right foot of the first A in AMERICA. On yours it is midway between the F and the A. There is really no reason to examine further, it isn't an 1804.
I agree with Condor, but another quick attribution is the 0 in the date is right where the 'O' in of is on the other side, which is diagnostic for 1804.
Now you have asked a much tougher question. I would want better, larger pictures, and oriented properly before I would take the time to try and determine what the coin actually is.
