Both A) and C) are grammatically correct, however, they have slightly different meanings.
had had is the past perfect of have
A simple version of the sentence without the past perfect:
Joshua had his car washed
had had his car washed puts it further in the past.
has had is also correct, because it's the present perfect tense of have
The present perfect tense tends to link something in the past with the present
So depending on the context, either A) or C) could be correct. They could be interpreted as meaning the same thing, but
had had technically implies something is further in the past than
has had does.
With more context the difference between past perfect and present perfect is more apparent:
Joshua had had his car washed at one point several years ago.
Joshua has had his car washed once a month for the last year.
In this example
had had suggests Joshua had his car washed once a long time ago. In the second example, it's clear Joshua has had his car washed once a month in the past, with the implication that he still does.
Both B) and D) have systemic verb usage/agreement issues