What are the differences between these ways to say "How?"?


  * 「どう」is used to ask what state something is in, right? 「今日の天気はどうですか」
  * 「どうやって」and 「どうして」are used with verbs to ask "how do you..." 「どうやってこの字を書きますか」, but 「どうして」is mostly avoided because it can also mean "Why", correct?

    1. Is there a way to say "how do you..." that is more polite than 「どうやって」and yet is not as ambiguous as 「どうして」？
    2. What do I use to say "How" as in, "to what extent", e.g. "How beautiful is she?" I am thinking it may be 「どれぐらい・どのぐらい」as 「くらい」has the meaning of extent, but I may be wrong. These two words also implicitly mean "How long" as well, right?
    3. Is there a way to say an emphatic how? E.g. "How is SHE beautiful!?" (In what way is she beautiful?)




>   * 「どう」is used to ask what state something is in, right? 「今日の天気はどうですか」
>

どう is the most basic word to ask "in what state" or "in what manner". You have
no problem to say どう書くんですか at all. どうやって is but a way to clarify you are
asking the method (literally, "doing what").

>   * 「どうやって」and 「どうして」are used with verbs to ask "how do you..."
> 「どうやってこの字を書きますか」, but 「どうして」is mostly avoided because it can also mean
> "Why", correct?
>

In modern Japanese, どうして is solely used for _why_ , or to say, _how come_.
Note that, however, other conjugations of どうする does not carry such meaning.
"How do you write this character?" is この字はどう（やって）書くんですか？

>   1. Is there a way to say "how do you..." that is more polite than
> 「どうやって」and yet is not as ambiguous as 「どうして」？
>

どうやって isn't particularly impolite, but relatively informal/colloquial. You
could use どのように（して） to sound formal, or いかに（して） to sound bookish.

>   2. What do I use to say "How" as in, "to what extent", e.g. "How beautiful
> is she?" I am thinking it may be 「どれぐらい・どのぐらい」as 「くらい」has the meaning of
> extent, but I may be wrong. These two words also implicitly mean "How long"
> as well, right?
>

どれぐらい・どのぐらい is the word to generally ask "what amount/degree", so whether it's
of time, space, percentage etc. is up to collocation. 彼女はどのぐらいきれいなんですか？ is
"how much", while どのぐらい待つんですか？ is "how long".

>   3. Is there a way to say an emphatic how? E.g. "How is SHE beautiful!?"
> (In what way is she beautiful?)
>

It's more like a rhetorical question (pun intended). If I'm correct that it
means "Tell me how she is beautiful while I could not find any way how she
is.", the Japanese way to say it is 彼女 **のどこが** きれいなんですか？

