No

Yes. The usual situation calls for the 2nd sentence.

Thank you for yoour reply

What if I change it up like this:

Do you have any apples?

I don't have any apples.

Do you have any apple?

I don't have any apple.

Still the same?

Only these:

Do you have any apples?

I don't have any apples.

Any pencils would suffice.

Any pencil would suffice.

I seem to use the second sentence a lot more than the first sentence.

Does the rule apply to this context?

No, that seems to be a different situation. In the first, the speaker is enquiring about the existence of apples; in the second, he is talking about the quality of the pencil.

So, it's

Mark: Hey, do you have any pencils? Any pencil will do.

Tom: Sorry, I don't have any pencils.

Am I right?

Yes, that sounds very natural, Michael.