1. Do you remember the day when we met first?

Question

1-1. Do you remember when we met first?

2. Do you remember the day on which when we dmet first?

(Can we use #1-1?)

3. Tell me the reason why you were late.
4. Tell me why you were late.
(We can delete 'the reason' in Sentence 3. Then we get #4. In #4, what is the part of speech of 'why'? In #3, 'why' was a relative adverb.' In #4, is 'why' a relative adverb or 'an interrogative adverb'?)

5. This is the reason why you were late.
6. This is why you were late.
(In #5, 'why' is a relative adverb. In #6, is 'why' a relative adverb or an interrogative adverb?)

Answer 1

1. Do you remember the day when we first met? <~~Word order matters!

1-1. Do you remember when we met first? not OK

2. Do you remember the day on which when we met first? not OK

Any of these can be used correctly if you put "first" before "met."
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3. Tell me the reason why you were late.
4. Tell me why you were late.
(We can delete 'the reason' in Sentence 3. Then we get #4. Yes. You can use either 3 or 4, but the redundancy makes 4 the better phrasing.

In 4 and 6, "why" is an interrogative adverb that is used to introduce the indirect question. I wouldn't use 3 or 5 because of the redundancy.