How can I find out if my copy of Black Beauty (Grosset and Dunlap) is an original?
It seems that you do not have a first edition, as the information I've found indicates that the first publication of Anna Sewell's novel was by London's Jarrold and Sons. (Grosset and Dunlap is an American publishing house.)
There is not an exact known publication date, though it is thought to have been originally published between 1877 and 1894.
While it is not a first edition, yours may be an early edition to Grosset and Dunlap's Young Readers, started in 1898 (G&D most famously published the "Nancy Drew" and Hardy Boys" novels, among others).
Condition is everything in the antiques book world. It is good that you find it in mint condition. "Mint" will mean that the cover is intact, (which will include no bent corners on the spine) there is no water damage, and the spine is not cracked. If this is the case, you may be able to fetch some money for your volume.