* 「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?)
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grammar word-choice words
What are the differences between these ways to say "How?"?
1. I dont think どうやって is particularly impolite, but you could use どう + verb mas stem + ばいいですか？ (literally how would it be good if I 'something'ed?) eg 報告はどうすればいいですか？ (how should i do my report?)

  2. how as in how much: どんなに「形容詞」（かのじょはどんなにきれいですか？）

  3. similar to 2 かのじょはなぜ **そんなに** 奇麗なの？？
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1. I don't think どうやって is impolite, but depending on the situation you might be able to use "どのように" or "どんな風に", which sound a little more formal/polite to me. (ex: どのように勉強すればいいんですか？)

  2. Your examples どのぐらい and どれぐらい are good (though you can also use くらい instead of ぐらい). You can also say どれだけ (abbreviated as どんだけ) or いかに (this sounds more literary to me), or (as another answer mentioned) どんなに. 

I wouldn't say that the first two words mean "how long", they are more general
than that. But in some contexts they can imply that.

  3. Rather than literally translating "How is SHE beautiful?" I would recommend a phrase like: 彼女が（本当に）そんなに綺麗なの？

Another answer used a は but I think が is very important here since the nuance
is "HER?!".

A related word that can add emphasis is to say あんなに ( _that_ much) instead of
そんなに.
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>   * 「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？
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