Learn English – Opposite of “Sugar Daddy”: A young person supporting an older person (financially, materially, sexually)

Green’s Dictionary of Slang dates its usage from the late 19th century; fair in the sense of justifiable:

[late 19C+] (orig. UK Und.):

• a justifiable arrest; usu. in the tongue-in-cheek phr. it’s a fair cop guvnor, put the bracelets on...

• any situation seen as fair and about which there is no complaint.

Wiktionary cites an early usage:

1891, Montagu Stephen Williams, Later Leaves: Being the Further Reminiscences of Montagu Williams, Q. C., Macmillan and Co.:

• "Several other witnesses gave corroborative evidence, and a constable who helped to arrest the prisoners stated that one of them, on being taken into custody, said: 'Ah, well, this is a fair cop.'"

Little Oxford Dictionary of Word Origins notes that:

The verb cop meaning to catch comes from northern English dialect cap meaning to capture or arrest. This probably goes back to Latin capere to take or seize. So a copper was a catcher which is why it became an informal term for a police officer in the 1840s.

Apprehended villains have been saying It’s a fair cop! since the 1880s

The Cambridge Dictionary has the word

grime

noun

a layer of dirt on skin or on a building:

This is typically used to mean ingrained dirt such as you describe, not fresh dirt, and does not easily wash off.