좋아하지 않아요 = I don't like it / them. (It could be "I don't particularly like it" or "I dislike it")
싫어해요 = I dislike/hate it. (It is clear and stronger about the negativity)
2. Yes. Both 좋아하지 않아요 and 안 좋아해요 are common.
3. Practically all verbs and adjectives can be used with both negative forms. For example, 어려워요(adj, native word) => 안 어려워요 / 어렵지 않아요; 실패해요(verb, hanja word) => 안 실패해요 / 실패하지 않아요.
One notable characteristic of the hanja(Chinese-originated characters) words like 실패하다 is that they can have two forms with the 안 adverb: 안 실패해요 and 실패 안 해요 (the latter form puts the negator 안 closer to the action suffix 하다, which makes it more forceful). With some verbs, both forms may be used, but with others (less common or more formal words), only the second form sounds natural (But this is not strictly for hanja words only - even native Korean 하다 words can sometimes behave this way: e.g. 좋아하지 않아 = 안 좋아해 = 좋아 안 해, although 안 좋아해 is much more common. You don't really need to know which 하다 words are hanja-based.).
A: 너희 학교 입학 신청 마감했어? (마감하다 is a native Korean word for stopping application)
B1: 아니, 아직 마감 안 했어. (안 마감했어 sounds not as natural)
B2: 아니, 아직 마감하지 않았어. (-지 않다 always has the same form. It is used less frequently in conversation)