Possible

I prefer these:

It is not possible that he did it.
It was not possible that he had done it.

Does the second one sound wrong to your native ear, GPY?

TakaDoes the second one sound wrong to your native ear, GPY?

It does not sound badly wrong, just not as good as the alternatives.

Then what about these pairs? Don't you think the second one sounds good?

The reason is that he worked hard
The reason was that he worked hard.

It is because he worked hard.
It was because he worked hard.

TakaThen what about these pairs? Don't you think the second one sounds good?The reason is that he worked hardThe reason was that he worked hard.It is because he worked hard.It was because he worked hard.

Yes, they're all OK.

And could you tell me what the difference in meaning is?

TakaAnd could you tell me what the difference in meaning is?

The versions with "is" are used when the thing referred to ("the reason" or "it") is still in existence or still relevant to the present situation. For historical events/situations, use "was".

Why is it impossible to apply such an interpretation to the first pair of "possible"? Isn't it possible to take the first one as implying that the speaker/writer is thinking about the possibility now, and the second one as implying he/she is seeing the past and thinking about the possibility then?

It isn't impossible to apply such an interpretation. (Remember that I said "It does not sound badly wrong".)

The difference between "It was not possible that he did it" and "The reason was that he worked hard" is that "worked hard" is more likely to be used in a sense of regular or continuing past activity. So, it is reasonable that "he worked hard" even at the time of "The reason

So would you come up with any examples where "It was possible that [simple past]" works fine and sounds natural?