I am not sure there is a hard and fast rule about the "not only... but also" construction, but my instinct when looking at your example sentences is that only #2 is completely natural without the "also" included, while 1, 3, 4, and 5 are various degrees of weird.
So if we examine what makes #2 different, I'd say that it's the fact that, in number two, the two ideas being contrasted – "sunny" and "raining" – are very close to direct opposites. If you read "not only when it was sunny," you can reasonably guess how the sentence will end, even if you're not totally sure which word it will be (rainy? cloudy? overcast? gloomy?).
Contrast that against #1, where we are comparing strawberries and bananas. First off, bananas are one of a zillion fruits; second, there's no such thing as opposite of strawberries (besides no strawberries); and finally, depending on the context, the second half of the sentence might not even need to be about another fruit at all. It could plausibly say "I like not only strawberries but also Ferraris." I'd posit that that's the reason #1 sounds not just unnatural but actually ungrammatical to me.
Sentence 3 suffers from a similar problem as #1 in that we can't really make a guess what the end of the sentence will be by reading only the first half because governments can be helpful for many different things.
Sentences 4 and 5 are somewhere in the middle in terms of naturalness. The fun vs educational dichotomy in #4 is common enough that that sentence sounds decent enough to me, but still just a bit weird. Similarly #5 I think is fairly sensible, but just slightly unnatural. I'd say both 4 and 5 are grammatical but unnatural.