Resolved questions
How do I say: "Happy New Year" in Japanese?

There are few ways to say, depend on the situation: 　　　

Colloquial:　明けましておめでとうございます＝Polite
　　　　　新年、明けましておめでとうございます＝Polite
　　　　　　明けましておめでとう！＝Casual
　　　　　　新年おめでとう！＝Casual

Writing:　　謹賀新年＝Formal

We say 正月(shougatsu)　and 元旦(gantan) to discribe in a word in Japanese.
If you wish a good year for someone you know, you cay say:
（今年も）よいお年を！　（This year)　I wish you a good year for you!
If you want to have a good relationship with someone through the year, you say:
今年もよろしくお願いいたします＝Polite
今年もよろしくお願いします＝Polite
今年もよろしくね＝Casual

Best of luck! and hope you a happy holiday!