Yes, rather can be used for "I would rather skate than walk", but it has another meaning. For example, "It was rather frightening" or "It was rather boring". It is usually used to describe something. Be careful though, this usage is rather formal :), so it's not used like this a lot in colloquial situations.

In the US, rather and quite are very strong intensifiers, often meaning very, totally, or really. Rather tends to be used with an adjective / adverb that is considered undesirable, and quite with an adjective / adverb that is considered to be desirable although this rule is not strict. "Quite beautiful / quite gorgeous, rather ugly / rather hideous. For the British use of rather and quite, Americans usually say pretty (oddly enough).

Your example is correct,but we also use this word to mean "a bit" or " a little", for example "I am feeling rather hungry today". I hope that is clear.

You'd have to say "I'd prefer skating rather than walking". I think this word has more meanings, but I don't know.