손발이 맞다.

Yes, it is most commonly used for two people, but it can also be used for a whole team or group.

Yes, it's a polite and positive way to describe your working relationship. Just use the formal ending: '{손|手}{발|足}이 잘 맞습니다.'

'호흡이 맞다' (breaths match) is slightly more formal and often used for artistic or high-level professional contexts, while '{손|手}{발|足}이 맞다' is more common in everyday tasks.

No, that sounds very strange and literal. Always keep the phrase '{손|手}{발|足}이 {맞다|合}' as a single unit.

Younger people often use '케미가 좋다' (chemistry is good) or '합이 좋다' (the 'hap' or match is good).

No, it applies to any collaborative task, including writing a report, planning an event, or playing a video game.

Use the form '{손|手}{발|足}을 맞춰 보고 있어요.'

Only if you are performing a task with the animal, like a dog and its trainer in an agility competition.

Yes, it is a standard Korean idiom used across the peninsula.

You can use it sarcastically, but usually, it's a positive phrase. For bad deeds, people often use '한통속이다' (to be in cahoots).

Yes, it comes from the same root meaning 'to fit' or 'to be right'.

Yes, if you are talking about how they manage their household or raise children together.