お帰りなさい probably comes from something like "よくご無事にお帰りなさいました" in old days (like Edo era). So it is expressing your relief that someone came back safely again. You can use it whether if it's home, or region, or any group/entity. If your classmate came back to the class from a year study abroad, you can also welcome her by saying "お帰りなさい”.

1.出して Casual 2.出してください Formal/Polite 3.お出しください Formal/Polite/Humble 1 & 2 can be used by teachers to students, because it's just simply polite form. But 3 is too polite and humble for such situation, and mostly used in business. Your translation is fine as above. All the sentenses sounds like direct quotations to me, because it comes with "と". If indirect, I would say "月曜に出すよう言われましたが", but I've also heard that Japanese doesn't have clear difference between direct/indirect speech.

グラムはミリグラムより（単位が）大きい、と言いますね。

悩み is a simple noun, but 悩みごと is like a verbal noun. (contraction of 悩んでいること) And it gives a nuance that it is ongoing or temporary issue. More used in daily conversation. 悩みごとでもあるの？ - Are you having any troubles or something?? (to your friend who's been depressed for a while) 悩みごとは全部片付いたのでもう大丈夫です。- Now I'm ok, cause all my concerns have been solved!

"彩る（いろどる）" means to give colours, so I think "色彩" is a mixed word of "colour" itself and the way/state of "colouring". 1.このドレスは色が美しい　 2.このドレスは色彩が美しい In the above examples, I would imagin No.2 dress is with more colours or shades in a very good balance, whereas No.1 dress can be just one tone.

Here you are! Hope this helps you. 1 私が落ち込んでいると、猫のキティはいつもそばにいてくれる。（My cat Kitty always stays near me whenever I'm feeling depressed.) 2 私のお母さんはいつも怒らないでいてくれる。（My mother always stays patient = does not scold me.) 3 レジをすませちゃうから、そこにいてくれる？（Will you stay there while I'll finish off with my bill?) 4 今日はお客様がくるから家にいてくれる？（We're having a guest today, so could you stay at home?) "いて（る）" can be translated into "stay" in most cases, I think. Jnnk's example can also be translated as "Thank you for staying in love with me".

Supposing this is a talk about your travel plan, the translation would be... "What can we find in the neighborhood where we are going to this time?" "Mawari" = neighborhood, surroundings

This is a kind of difficult request, because greeting messages are often cliche and literal translation would sound not natural. But anyway, let me try. あなたにとって実り多い一年になりますように。 あなたがXの一員になれたことをとてもうれしく思っています。 どうぞお身体には気をつけて、ずっと幸せでいてください。 いつも応援しています。 Actually, the first sentence would fit better in the end, I think.