No.
It is likely that Mrs. Hamilton abused the slave Mary due to a combination of factors, including a belief in white superiority and power dynamics inherent in the institution of slavery. Mrs. Hamilton may have also been influenced by societal norms that condoned the mistreatment of slaves at the time.
Abolitionists
Yes, historically the name Clay has been associated with enslaved individuals in the Southern United States. It was common for enslaved people to be given a surname that matched their occupation or the material they worked with, such as Clay. Today, the name is used in many different contexts and may not hold the same connotations.
Not in the USA because to be a slave you had to have black blood in your system which means that if you had black blood in your system you were not considered human but cattle and therefore a slave. However outside America there were plenty of black slave traders. And black tribesmen involved in selling other black people into slavery. The whites were the end users; the blacks and the Arabs were the suppliers. This is not true at all. William Ellison was a Black slave owner in South Carolina and the majority of slave owners in New Orleans were Black. William Ellison, whose slave name was April, became a slave magnate and expanded his business throughout the south, from South Carolina to the Mississippi River. I would strongly suggest you Google "Black Slave Owners" and I guarantee that you will have a plethora of information.
A slave owner may want a slave who had lost hope because they would be less likely to resist or attempt to escape. A slave who has lost hope may also be easier to control and manipulate, making them more compliant and submissive. Additionally, a slave with no hope may be seen as less of a threat to the slave owner's authority and power.
Bernie Hamilton's birth name is Bernard Hamilton.
Chico Hamilton's birth name is Hamilton, Foreststorn.
Hamilton Revelle's birth name is Arthur Hamilton Engstrom.
Hamilton Ridenour's birth name is Hamilton Franklin Ridenour.
Hamilton Benz's birth name is George Hamilton Benz.
Hamilton Mosmann's birth name is Hamilton Mosmann Rodrigues.
Aidan Hamilton's birth name is Aidan John Hamilton.
Bevin Hamilton's birth name is Bevin Wynne Hamilton.
Carrie Hamilton's birth name is Hamilton, Carrie Louise.
Darryl Hamilton's birth name is Darryl Quinn Hamilton.
Iain Hamilton's birth name is Iain Ellis Hamilton.
Judd Hamilton's birth name is Judd Douglas Hamilton.