Personally, I feel both are strange. The sky is overcast. We use 'looks' to describe how something seems when we are not sure how it actually is. Or when we want to say that something has an appearance similar to another thing. This probably isn't the case here. The sky just is overcast. Or maybe there has been a huge fire and the air is full of smoke - the sky looks overcast (but isn't really). The weather is cloudy is also not very natural. We don't say the weather is hot, raining, windy. We say It is hot, rainy, windy, cloudy.

Yes, I would call these interchangeable.

Yes, you can, since "cloudy" and "overcast" are synonyms. That being said, I think there's room for nuance. The sky can be more or less "cloudy", making the sky behind it more or less visible. I would say "overcast" sounds more negative/pessimistic than "cloudy", as it places more emphasis on the idea of gloominess.

Yes, more or less