Sometimes only the context can help us decide if выше is an adjective or an adverb.
I agree with @Sergey Slepov that выше in sentences 5, 6, 8 can be only adverbs. Still, one can find such contexts that sentences 1, 2, 3, and 4 can be interpreted as having выше as an adverb, too.
For example, sentence 1, if we talk about mountain climbing, then imagine a photo of a mountain with several students climbing it, each at a different height. The context for выше being an adverb here is as following:
[— А где на этом фото Джек?]
— Джек выше других учеников в классе.
Sentence 2 can be looked at as describing that Jack climbed higher than somebody else:
- [Все ученики поднимались высоко на эту гору, но никто не дошёл до вершины.] Джек был выше других учеников в классе. [Он поднялся выше всех.]
Sentence 3 can also be about a photo with mountaineers, when two people are arguing about where exactly in the photo Jack is:
[— Вот это Джек, возле этого камня.
— Нет, это Стив.] А Джек всё же выше.
Sentence 4 can be a part of an argument about who climbed higher, Steve or Jack:
[— Стив был на высоте 3,5 километра.]
— А Джек был всё же выше.
Only sentence 7 has выше as an adjective without a doubt, since мелкие склоки и личные амбиции are abstract notions, one cannot be physically above or below them. Sentences 1—4 can have either an adjective (most likely) or an adverb, sentences 5, 6, and 8 have adverbs.