Both きける and きこえる fill the role of "to be able to hear," but an important difference is that きける is "able to hear" in the sense of "if you turn on the radio you will be able to hear the news." (The option is there to do it if you choose.) Whereas with きこえる, only the physical perception is being talked about. Either you can physically hear something or you cannot, there isn't really a choice in the matter. 見られる (e.g. you can see animals at the zoo [if you go there]) and 見える (can see the airplane [if you squint]) work similarly. Let's go down the list:
(1) 雨が聞こえる。
(2) テレビを聞く。
1) can hear the rain (the sound waves penetrate one's ears. Potential verbs generally take が). 雨の音が聞こえる。
2) "to listen to the TV" (plain statement)
(3) あなたの意見が聞きたいです。
(4) 耳がよく聞こえません。
3) "I'd like to hear/ask your opinion." (don't forget that 聞く can also mean "to ask"!)
4) I can't hear well. (physical limitation)
(5) あなたの意見を聞きたいです。
(6) 耳がよく聞きません。
5) This isn't strictly incorrect as there are cases where you can use を with potential verbs/verbs of desire, but generally speaking one should use が. The たい forms should be treated as adjectives instead of verbs.
6) This doesn't work because you're talking about the physical perception of hearing, not the circumstances that allow you to hear something.
Hope that clarifies things a little!