a) is correct. The pattern of that sentence is
[clause in simple past tense] ["after" clause in simple past tense]
This pattern is really really common in English.
b) is incorrect. The pattern of that sentence is
[clause in present perfect tense] ["after" clause in simple past tense]
This pattern is never used.
A present perfect tense clause is not used with an "after" clause. An "after" clause tells you a specific event occurs at a later time. Whenever you have the pattern
[clause a] after [clause b]
[clause a] should be a description of a specific event like "John ate an apple". The present perfect tense is not describing a specific event, it is merely a statement saying that an event existed. It's a subtle difference, but basically:
A simple past tense clause is a concrete action. If someone said,
Please draw a picture of "John ate an apple"
you would draw a picture of a man eating an apple, because the sentence is a concrete description. But if someone asked you
Please draw a picture of "John has eaten an apple (in his lifetime)"
you would draw nothing, because the sentence is just an abstract fact.
On the other hand, you could use the pattern
[clause in present perfect tense] ["since" clause in simple past tense]
Because the word "since" is used with non-specific events. It makes sense, because "since B" tells you about the all the moments after B. Whereas "after B" tell you about a specific moment after B.
So you could ask
Have you washed your hands since you came home?
c) Wrong for the same reasons as b)