In the sentence "Something has gone wrong", the subject is something and the predicate is has gone wrong. The predicate can in turn be broken down into the verb has gone (which is the present perfect of go) and the adverb wrong.
You can tell that wrong is part of the predicate, and not a sentence modifier, by the fact that you can't move it to the beginning of the sentence: *"Wrong, something has gone" would not be grammatical. So we know that wrong is merely modifying the verb go.
In "The sun was shining bright and warm", the situation is the same: the subject is the sun, and the predicate is was shining bright and warm (where was shining is the verb and bright and warm is an adjective phrase, or — according to some authorities — a kind of adverb phrase).
It's confusing to me that in the sentence "she looks happy" the predicate is "looks happy". I would put "looks" as predicate and "happy" (looks how?- happy) as an adverbial modifier of manner, as in sentence "He speaks English fluently", "fluently" is an adverbial modifier.
There are two problems with your reasoning here.
The first problem is that in the sentence "He speaks English fluently", you seem to be under the impression that the adverbial modifier fluently is not part of the predicate? But it is part of the predicate. The subject is he, and the predicate is speaks English fluently, where speaks is the verb, English is the direct object, and fluently is an adverb modifying speaks English.
The second problem is that in the sentence "She looks happy", the word happy is actually an adjective modifying she, rather than an adverb modifying looks. What's more, it's a complement, rather than an adjunct. (The verb look is one of a handful of verbs whose complement can be an adjective modifying the subject. Other such verbs include seem and sound: "He seemed annoyed", "You sound tired".) So "She looks happy" actually has quite a different structure than "He speaks English fluently", though in both cases the first word is the subject and the rest is the predicate.
In the sentence "For him to leave the town in early spring was an unbearable thing", the subject is "For him to leave the town in early spring", and the predicate is "was an unbearable thing". That subject is what's called a small clause or infinitival clause: it contains an infinitive phrase, "to leave the town in early spring", with a subject, "him". "The town" is the direct object of "to leave", and "in early spring" is an adverbial modifying "to leave the town".