Does anyone know the origins of “clerical work”? Is it related to the idea that only clerics (well, monks) at one time were the only ones who could write and therefore jobs involving writing or copying became “clerical work”?

Yes. But “clerics” didn’t just include monks; it also included secular priests, people in minor orders such as lectors and deacons, and choirboys (the Children of the Chapel Royal were referred to as “the little clerks”) whom we would not now think of as “clergy”. The link between literacy and clerical status was so strong that the ability to read was taken as a valid proof that a person was in holy orders and was entitled to Benefit of Clergy - the right to be tried in a religious court, which passed more lenient sentences than the secular justice system.

Exactly.

The original sense of clerk is cleric. But the sense of someone who performs scribal duties is almost as old.

The adjective clerical appears in the late 16th century meaning something pertaining to the clergy. The modern sense appears at the end of the 18th century.