Why whales head south [for the winter]
1.Why wales head south for the winter
2.Why wales head south in the winter
3.Why wales head south during the winter

Question1. #1~3 are fine?
They can be used in the case of meaning for a non-specific winter? What do you think of it?
In my thought #1~#2 are fine. #3 isn't fine because it's not a specific winter.

Question2. the winter or winter
I found that native speakers uses both 'the winter' and just 'winter' right?(even though it's not the specific winter)

4.The house can be very cold in (the) winter. -from OED

So I can use any of them, right?

Question

Why whales head south [for the winter]
1.Why wales head south for the winter
2.Why wales head south in the winter
3.Why wales head south during the winter

Question1. #1~3 are fine?
They can be used in the case of meaning for a non-specific winter? What do you think of it?
In my thought #1~#2 are fine. #3 isn't fine because it's not a specific winter.

Question2. the winter or winter 
I found that native speakers uses both 'the winter' and just 'winter' right?(even though it's not the specific winter)

4.The house can be very cold in (the) winter. -from OED

So I can use any of them, right?

Answer

The use of the definite article before the seasons is possible even when we are speaking in general terms — it doesn’t affect the meaning. “In winter” or “in the winter”. In your examples, I like to use the article, but it’s perfectly correct either way. On the other hand, I usually say “it’s winter now”, but it’s also perfectly correct to say “it’s the winter now”. If we want to refer to a definite winter, then we have to use the article: “In the winter of 2017”. Names of seasons can be capitalized, but it’s optional (a matter of style).

“Whales” is spelled with an H after the W. In parts of the UK (mainly Scotland) and the US (mainly Texas), the combination is still pronounced as an unvoiced or aspirated W or similar. “Wales”, on the other hand, is a region in the UK.

Now to the main point: In the phrase “for the winter”, “for” means something like “in order to”, so the phrase means “in order to spend / pass the winter”. It’s my understanding that the whales make the trip *before* the winter, not during / in the winter.

Also, just to be thorough, let’s point out that none of your phrases are questions or even sentences — they are subordinate clauses. That’s OK, as long as you know in what context to use them. You could say “I’ll tell you why whales go south for the winter.” Alternatively, you could ask a question: “Why do whales go south for the winter?”

Answer

Good.

Answer

Cally An is right.
Yes, non-specific winter.
.
"for the winter" means they are gone by winter.
heading South "in the winter" or "during winter" would mean that the journey is in winter. Slight difference in meaning.
Lose "the" before winter for the during one.
Ducks fly North in winter. (Southern hemisphere perspective :) )

Answer

The grammatical structure for all three phrases is correct. However, if you want to make the sentence/question proper it should be: Why do whales head south during the winter?
Adding "do" makes it a proper question.
If it's a sentence, it may be more correct to say something like:
"The reason why whales head/go south in/during/for the winter is because..."
Using either "winter" or "the winter" is alright. However, if you are using "for" before "winter" saying "for the winter" may be better. :)