I wish + past subjunctive = describing an unreal world about the present. Use this where you would use the present if the world were real.
(by "past subjunctive" I mean the past tense for every verb but "to be" and either "was" [informal] or "were" [formal] with "be.")
You have two cars --> I wish you had three cars.
I am short--> I wish I was taller [informal]. I wish I were taller [formal].
I wish + past subjunctive progressive = same, but replaces present progressive.
It is raining--> I wish it wasn't raining [informal]. I wish it weren't raining [formal].
Use the conditional instead of the present after "I wish" to talk about something unreal that you want to become real in the future.
It is raining. I wish it would stop.
You are going too fast--> I wish you would slow down.
This is why "would" sounds better in your examples. When you are politely requesting your friend to turn down the record player, you're not comparing an unreal present world to the real present world. You're talking about something you wish to become real in the future. This also explains why "would" is better than the past in the first two sentences about the rain.
For the last two, I don't like "I wish it stopped raining now" even if you are talking about a hypothetical world in the present. I think this is specific to the verb "stop" and the "stop --ing" construction, but I can't think of a rule -- perhaps someone can edit one into this answer. At any rate, the most natural way to say this is either
I wish it would stop raining now.
I wish it weren't raining now.
if you insist on being hypothetical.
EDIT: Upon further reflection, we use "wish x stopped v-ing" to refer to habitual action and not to present action. In other words, it replaces a present, but not a present progressive:
He interrupts me all the time --> I wish he stopped interrupting me all the time.
It rains every Thursday--> I wish it stopped raining every Thursday.
Since "it's raining right now" it present progressive, it can't be replaced in this way:
It is raining right now -/->
I wish it stopped raining right now