Change in indirect speech

With no extra info, you should go with the second option:

He said that the teacher usually did not ask any question.

Only universal truths, e.g. facts of science and maxims, are forever true. Hence, we write these in the present tense in reported speech.

"Usually" can be used in a past-tense sentence when something was known to happen most of the times in the past. The event may or may not happen anymore.

Direct speech

He said, "The teacher usually [does not] ask any question."

In direct speech, the above is correct.

Indirect speech

Indirect speech is normally used to refer to the past: we change does from its 3rd person simple present to its 3rd person simple past and we introduce the quoted speech with a relative pronoun that (although this can be omitted) to make it indirect. However if:

b) there is no change in tense

How do we known that the speech is direct or indirect without speech marks?

Take for example the following sentences:

1. He said the teacher usually does not ask any question.

2. He said the teacher usually did not ask any question.

In the above we can say that both 1 and 2 are both in indirect speech. However in terms of tenses, we know that the first, uses the tense for direct speech and the second, uses the tense for indirect speech.

Now, let's add the correct punctuation:

1. He said, "The teacher usually does not ask any question."

2. He said the teacher usually did not ask any question.

Here 1 is clearly direct speech, while 2 is clearly indirect speech. Without a distinction in tense 1. can become indirect if the speech marks and comma were omitted and 2. can become direct speech if you include speech marks after the and question.

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To answer your question, the second option is the safest option so that there are parameters for direct vs indirect speech.

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Any question vs any questions

"any question" may be idiomatic, but is typically unheard of. However, ykombinator advises that if any question:-

[is] at the beginning of a sentence, in a more certain, assertive, authoritative tense, singularity is fine.

Although, notes, the "plural sounds better".

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