"The weather is a little bad today."
ちょっと modifies the whole sentence, neither 天気 nor 悪い. Oriental languages cannot be always fully analysed in the form of subjects, verbs, objects, modifying structures etc. This sentence is an example. In this sentence, 天気が悪い can be viewed as a property of today.
In teaching / research of Japanese grammar, a typical example is 象は鼻が長い。象 [elephant] is the topic, 鼻[nose]が長い is a clause. Though it could mean 'The elephant's nose is long', it actually means something like 'The elephant is (some kind of animal) whose nose is long. ', analysed as ( 象は [ (鼻が) (長い) ] ).
In the same way, the sentence in question is ( (今日は) (ちょっと) [(天気が) (わるい)] (です) (ね))。
ね is another final particle. It is used here because the speaker wants the listener to agree what the sentence states. Perhaps the sentence can mean 'Today the weather is a little bad, right? / em?' etc.
Is the verb part of the clause or the main sentence?
Because your breakdown of the sentence in question seems to imply that です is part of the main sentence, but that would contradict the interpretation of が as indicating grammatical subject.
Also, does ちょっと always have to be part of the main sentence or would it be possible to put it after が and let it be part of the clause and only modify 悪い?
To me it would seem that です does in fact fall into the 天気が悪い clause, because the は indicates the topic rather than serving a grammatical function, almost like when we might say "this food, do you like it?" This food in the sentence doesn't really have a place, but it is stated because it is necessary to know what we're talking about.
But honestly, I wouldn't worry about it a ton, because as the previous person said, thinking in terms of subjects and topics as it relates to grammar (at least, grammar as westerners are familiar with) is not always correct or even helpful when trying to understand the mindset and culture that utilizes said sentence structure.
I think the reason we feel uncomfortable about that ね is that so far, in every Japanese to English translation with a ね at the end "isn't it" has been required, so duolingo has conditioned us to expect that. Same with よ at the end. "You know" is the one and only required translation, even though in real life it may be better translated with a "very" or just a "!"!
No idea dude. But I´ve been looking around a bit and most translation engines make this sentence into: "Today is a bit bad weather" which is nonsense. While if you remove the ね it becomes: The weather is a bit bad today. Which actually makes a bit of sense. But if it was left to me I´d just go with the easy version: 今日は天気が悪いです The weather is bad today. Might even leave out "today" since if you´re already talking to someone... Well, they´d know you´re talking about today.
The ちょっと could be before 天気 because, if you think about it, if the weather they're having is only a little bad, that means you could say they're just having a little weather. Not much is going on with the weather, so we're just having a little weather. So you could think of it as, "The little amount of weather that we have is bad."