The core meaning of the verb けす is not "turn off" but is "put off (a fire)" or "extinguish." The gas stove is a recent invention (I believe from the 19th century) and before that, people used to say things like "たきび の ひ を けして ください" and "かまど の ひ を けしてください (=please put out the fire in the oven/furnace)," and you still here "たばこのひをけしてください” and “ろうそくのひをけしてください” So the expression in question doesn't actually mean "please turn off the fire of the gas." It's more like "please put off/extinguish the fire on the gas (stove)." If you think this is still too verbose, you can trim it into "ひをけしてください" by dropping the “gas” part, not the “fire” part.

First, the golden rule for second-person pronouns in Japanese: avoid using them. It's not a matter of choice between anata, kimi and omae. Instead, simply don’t use any of these and say things like “どこにお住まいですか (where do you live?)” and this is the most natural sounding way of saying things in 95-99% of the time. If you think this might confuse the listener, you can always start your sentence with his or her name or title (“田中さん、昼ご飯は食べましたか？”) Having said that, you are on the right track. Omae is mostly guy talk (variably translated as “dude” (friendly), “you bastard” (battle talk) and most of the time,…. simply “you”) but some girls might occasionally use it in very informal situations for dramatic effect, especially in mock exasperation or confrontation (that is, as jokes). For example, a girl is narrating a story to a close friend and she says “….and I felt like he was trying my patience and I was like ‘Look who’s talking?’…” (なんか私もう我慢の限界で、「おまえがよく言うよ」って思ったんだけど、・・・。) In this case, she might choose to invoke omae to show how annoyed she was. But it's a bit like cursing so you should be careful with it.

"I suppose s/he was trying to figure out who s/he really was (in an effort to form an identity) and ended up hitting a brick wall."

BilingualになるよりBiculturalになるほうがさらに難しいといわれる所以ですね。僕も違う文化で育った仲の良い友達とはこういう話を良くします。自分と同じ町で生まれ育った人とですら、いろんな原因で怒らせてしまったりこちらがむかついたりすることも珍しくないわけですから、文化が違うとなおさら色々あります。さらには、人々の振るまいとか考え方が違うのは、さらには宗教や政治体制などの難しい問題も関係していて、本当に難しいと思います。だからこそ、それを乗り越えて理解できたときの充実感は格別ですね。 ただ、実は文化というのは国や文化だけの問題でもありません。どの国でも大都市には大都市に共通の特徴があったり、田舎には田舎に共通の特徴があったりしますし、エンジニアはどの国の人もエンジニアらしい発想をしますし、どこの国の弁護士も似たようなしゃべり方をします。例えば、スウェーデンの弁護士をアメスウェーデンのロックシンガーと日本の弁護士の二種類の人間と比べるとすると、恐らくスウェーデンの弁護士は自国のロックシンガーより、日本の弁護士のほうに、色々な意味でより行動や思考が似てるとも言えると思います。そういう発見もまた面白いです。

Another toughly… The difference can sometimes be very subtle but here's my take on it. 「はじめ」は単純に期間の最初の部分を意味する(a time from which something begins/the first part of a period of time)のに対して、 「はじまり」は何か大きな一連のイベントの最初のきっかけ(an event that prompts a chain of events)、または人生や歴史上などのイベントや物語の最初の部分(the first part of a significant/historical event or a story)を指すことが多いと思います。英語ならoriginやopeningと訳せる場合もあると思います。 Going back to your example, the person in question has been whiling away his vacation and the 夏休みのはじめから part is there to specify since when, which means the speaker obviously doesn't have an exciting story to tell about this summer vacation. To generalize a bit, 夏休み is usually seen as just a period of time that will start automatically every year without some first cause setting it in motion as long as you are a student or an employee at your organization. For this reason, it collocates better with はじめ just like "今週のはじめ" and "今年のはじめ". To be sure, expressions like 楽しい夏休みの始まり are also possible and, in fact, not so uncommon, but they give you the impression that the speaker thinks of summer vacation not as a span of time, but as a whole series of important events (she might have gone on a month-long trip to some nice place and have a story to tell). More examples： 今週のはじめからずっと風邪を引いている。 ４月から会社に勤め出しました。はじめは慣れませんでしたが、いまは楽しくやっています。 事の始まりは、３年前日本に旅行した時のことです。=It all started 3 years ago when I took a trip to Japan. 宇宙の始まり=the origins of the universe そもそもけんかの始まりは何なの？=So what started the fight in the first place? あの小説は始まりのシーンがとてもいい。=I really like the opening of that novel.

これからプレゼンテーションの準備にとりかかるところですが、特に興味をお持ちの機能などはございますでしょうか。ご連絡いただければ、プレゼンテーションに含めるようにいたします。 In formal settings you are not supposed to literally ask what the other person wants to do, because it would sound a little too direct and nosy about their intentions. Instead you can rephrase it and inquire about their interests, plans, concerns and the like. Even in English, phrases like "Would you like to...?" and " "Would you be interested in..?" sound more polite than "Do you want to..?" except “..したいですか” comes across much ruder and snoopier than "Do you want to.." does in English.

Use と for quoting the speech itself. ex 天気予報で明日は雨だと言ってたよ。 お母さんは冷蔵庫には卵がもう無いといってたけど、さっき冷蔵庫を開けてみたらまだたくさん残ってた。 Use を if you just want to specify the type of utterance (lies, complaints, the truth and so on) ex 嘘を言う。悪口を言う、文句を言う、本当の事を言う、文章の内容を言う。 実を言うと、昨日の夜は寝られませんでした。

This is a tough one but I would suggest avoiding too many のs: 20世紀初頭の中国および日本の知識人に見られる地域主義(思想) 20世紀初頭の中国および日本の知識階級における地域主義(思想) 20世紀初頭の日中知識人の地域主義(思想)