Question

1. I know why she was late.
2. I wonder why she was late for school.
3. I doubt why she came back home late.
4. He told me how he got there.
5. He asked me why I didn't study.
6. He showed me how to get to the station.
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Are they all grammatical? Are both 'why' and'how' interrogative adverbs in the sentences?
Writeacher
All are grammatically correct, but #3 seems awkward. For that one, it'd be better to write/say, "I doubt the reason she gave for coming home late."

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