One (very slight) difference is that 'would rather' is often used after someone has suggested the opposite. For example, consider the following conversation: A: Would you like to join me for a drink? B: I would rather be alone, if you don't mind. Whereas, 'prefer' is often used to indicate preference without any negative connotations. Consider: A: Why are you taking this job? B: I prefer to work in the countryside.

Yes, they are absolutely interchangeable in these sentences. In fact, they are almost always interchangeable. I can't really think of any examples where they are not interchangeable, to be honest. :)

All of your examples are correct. However, when using "would rather", a comparison is usually being made, such as in your final example. I would rather "this", than "that". While it is most often stated in the same sentence, it is also acceptable to use it alone when the prior context of the conversation will make it understood.

In the sentences above, yes, "would rather" and "prefer" are interchangeable. When you use "rather" without "would", it takes on a different meaning. "He is a rather handsome man" means that he is very handsome.

There is a slightly different feel depending which one you use and native speakers use them whenever they feel like using either of them. You can use them interchangeably for now and perhaps if you are surrouned by native speakers in the future you will see the slight difference yourself. Many people say that there is no difference for them though. And yes all of these sentences sound perfectly fine and don't heistate to use them whenever you want ;) Nevertheless you have to remember that after "would rather" you use different grammar patterns and different ones after "prefer". There's also "would prefer" which is basically the same as would rather. And if you really want to know this slight difference then here it is "you use prefer when you want to let someone know that the thing you prefer is something you like in general (not just now) but with "would rather/prefer" you indicate that it would suit your liking more to do something in a particular way when talking about something now, something that won't last long. Yet still people tend to use them as they want.