1 is not common or particularly idiomatic; there are no Google hits for the question outside of Stack Exchange; but it doesn't seem ungrammatical. To me, 2, 3, and 4 seem worse.
It relates to which verb is most natural to use for describing something.
The building has three stories.
The tent holds three people. (or accommodates, is big enough for, etc.)
The processor has three cores. (or contains, etc.)
So you'd ask
How many stories does the building have?
How many people does the tent hold?
How many cores does the processor have?
In contrast, you'd say
It's three times something else
Or you could say (although I don't think it's very elegant, and people often get confused by statements like this):
It's a three-times increase (or three times increase, or three-fold increase)
So, while it's uncommon, you could say
How many times an increase is it? (or was it)
But I wouldn't recommend it. As the answer you link says, it's not a particularly elegant way to ask. People often get confused whether "increased three times" means by an additional three (hence four times bigger than the original) or three times bigger. You would do better to ask something like:
How big an increase was it?
Or
What did it increase by?
And then someone can explain it as clearly as possible.