I'm neither a native speaker nor a very advanced learner, though I think that it works like that:
한국어를 해요 - to speak Korean 한국어로 말해요 - to say/speak in korean.
This is because of '로' particle describing the way of doing something (Q: how are you speaking?A: In Korean)
I think most people would generally agree, and it's hard to say whether they actually want to depend on this level of nuance, but "speak Korean" and "speak in Korean" are slightly different.
Adding a subject helps to illustrate the difference: "We speak Korean" serves less to describe an action, and more to describe an ability of the subjects. Whereas "We speak in Korean" is more of a momentary thing, that describes particular actions.
That said, I also agree that most people don't care about this level of nuance, so they should probably just accept both
어떨 때 means “in what kind of situation”, “in what kind of times” or even just “when”. However, 어떨 때 can also be used to mean “sometimes”. But this is when you want to show contrast between “some times” and “other times”. Therefore, you usually use 어떨 때 with the topic marker, -는.