Int. J 1-4 泊めてもらう・お互い

For today's studies I had to read a decently sized fake journal by a Japanese student visiting America. It was all pretty intuitive to understand but I'd love clarification on a few things.

１）になってしまった This was at the end of a sentence expressing that 'because of the international date line, the day I arrived in America was the same day I left in Japan'. To express that it was the same time, though, I believe the book stated 同じ時間になってしまった。 It's the translation of natte shimatta that throws me a little. Is it akin to the English, 'it ended up being the same time'?

２）泊めてもらう - The speaker is staying at someone's house til he can find an apartment. Is it akin to 'let me stay' as in a favor?

３）とても丁寧な日本語で自己紹介をされて、びっくりした。"[He] gave a self-introduction in very polite Japanese, and I was surprised." I want to know why されて and not して. Is it

４）お互い - All I desire is a little explanation of this word and its usage. I hear it frequently enough. 'Each other'?

Also, I'd love verification if I'm using these constructs right.

５）〜て初めてのN = The First N after V-ing
example: あのクラスに入って初めての難しい宿題でした。It was the first difficult homework after joining the class.

６）〜て初めてV。 = 'When V happens, then V happens for the first time."
example: スペインへ旅行して初めてスペイン語を普通の会話で話してみました。 I tried speaking Spanish in normal conversation for the first time when I traveled to Spain.

よろしくお願いします。

1)
～てしまった has a nuance of "unexpectedly / with regret".

2)
Right. I receive a favor of their letting me stay with them.

3)
The passive is used there because the subject/narrator is the writer. You can unify the subjects of the two clauses by using the passive form.

active
彼はとても丁寧な日本語で（私に）自己紹介をした。
passive
私はとても丁寧な日本語で（彼に）自己紹介をされた。

（私は）とても丁寧な日本語で自己紹介をされて、びっくりした。
(The subjects of 自己紹介をされて and びっくりした are both "I".)

4)
Yes.

6)
Unlike ～たとき初めて, ～て初めて is not just "It's the first time ~ when..." It connotes "finally".
e.g.
スペインへ旅行したとき、初めてスペイン語を話してみました。
I tried speaking Spanish for the first time when I traveled to Spain.

スペインへ旅行して初めて自分のスペイン語が通じないと気づきました。
(I had thought I could communicate with people in Spanish, but) I (finally) realized I couldn't communicate with people in Spanish for the first time when I traveled to Spain.

ledojaeger san,

A little more on 4) otagai

It does mean "each other" as you say, but the usage is fairly nuanced in practical language. You can also think of it as "mutually" or "reciprocally." In other usages, it's "both you and I" or even "in the same."

Some google-fu shows:

otagai daisuki desu (we love/like each other)
otagai no kaisha de (in each other's company...but natural English would be "in the same company")
otagai no tame ni naru (for the sake/benefit of each other)
otagai da/desu (the same, in the same boat)

An example of the last one:
Person 1: Shigoto ga isogashikute, neru jikan ga nai. (I'm so busy at work, I don't have time to sleep.)
Person 2: Otagai da. (You and me both.)

There's also the phrase otagaisama (same here/in the same boat). It's similar to the example just above.
Man: Kekkon shinakute yokatta (I'm glad we never got married!)
Woman: Otagaisama da wa (You and me both, buddy! or Same here!)

Person 1: Kachou ni okorarete, toubun zangyou da (I got yelled at by the section chief; I'll be doing overtime for the next while)
Person 2: Otagai sama da (Me, too/I'm in the same boat)

[a native can help here if I missed the nuance, but I think it's pretty solid]

butarox said:

otagai no kaisha de (in each other's company...but natural English would be "in the same company")

Click to expand...

In their respective companies, I think would be the acceptable English translation.

butarox said:

An example of the last one:
Person 1: Shigoto ga isogashikute, neru jikan ga nai. (I'm so busy at work, I don't have time to sleep.)
Person 2: Otagai da. (You and me both.)

There's also the phrase otagaisama (same here/in the same boat). It's similar to the example just above.
Man: Kekkon shinakute yokatta (I'm glad we never got married!)
Woman: Otagaisama da wa (You and me both, buddy! or Same here!)

Person 1: Kachou ni okorarete, toubun zangyou da (I got yelled at by the section chief; I'll be doing overtime for the next while)
Person 2: Otagai sama da (Me, too/I'm in the same boat)

Click to expand...

お互いだ is hardly used as the answer in the first example. お互い様だ can be an answer, but you need to pay attention that this phrase has a nuance of "I, too, am very busy/have to do overtime work, not only you, so you should bear it without complaint" in those contexts. If you just agree with him, 俺もだよ is the best choice.
See the following example.
手伝ってくれてありがとう。お前がいなきゃこんなに早く残業終わらなかったよ。
Thanks for your help. My overtime work must not be finished so soon like this if you wouldn't help me.
お互い様だよ。（＝困っているときはお互い様だよ。）
We can help each other. (= If I'm in trouble, I believe you must do the same thing.)

お互い様 works well when answering to the words of gratitude or apology.

Toritoribe,
Thank you for help. Also, Majestic and Butarox.
In your last example of お互い様, Toritoribe, the 'We can help each other' and its nuance. Is an accurate English way of thinking about it, 'You would have done the same for me!' answering to gratitude?

Yeah, that's right.

Well...looks like I got properly spanked on this thread... . BUT, I could swear that in real life (granted...I don't live full-time in Japan) I hear otagaisama in more negative situations than positive ones.