"The sky is cloudy."
I'm a little bit confused. This whole time, I thought it was an adjective _ cloudy I've looked this up briefly online, but I'm still having trouble grasping the concept because to me, 今日はくもりです now looks like "today is clouds", which is obviously not natural in English. 曇り is more like "cloudiness"? Is this a noun that somehow behaves like an adjective? If so, is there a particular term for these kinds of words? Are there other situations where this is commonly encountered? Or does です imply "has" in this case? Sorry for so many questions! Just trying to wrap my mind around it.
くもり is like "clouldy weather" and cloud is "くも". Just bear in mind that an adjective in one language does not necessarily translate into an adjective in another. If we want to say cloudy, probably the stateful verb くもっている is a translation that fits better, but in day-to-day くもり(noun) is as good as くもっている(stateful verb). I can't think of an adjective in Japanese that represents "cloudy" in English.
It's a different thing. Think of it as ~ています. It's present continuous, about the same as ~ing in English. In much the same way as the word sing isn't continuous just because it ends in ~ing, continuous things in Japanese aren't animate just because they end in ~います.
Also, things that end in ではありません aren't inanimate just because there's an ありません in there. It's just the negative form of です.