カレンダー (karendaa) or 暦 (koyomi) are the Japanese words for "Calendar".

The Japanese word for "calendar" comes from the English, so it is very close: 'karendaa.'

No. There is the Gregorian calendar (used by Americans, Europeans, and Japanese), a Hebrew calendar, and a Muslim calendar at least. Here's a link to a Chinese calendar and an Indian calendar.

The government held a monopoly on calendar printing.

The primary difference is in how they count years. They base their years on the reign of an emperor rather than the method of defining them by the Christian calendar.