Yes, it is an important part of Jewish life. It is also one of the 2 official calendars of Israel.
The Hebrew calendar is used in Israel, and by Jews all over the world.
The Hebrew calendar wasn't destroyed, we use it to this day. Today's date is: 14th of Elul, 5771 [September 13, 2011].
The Gregorian Calendar is solar and the Hebrew Calendar is lunisolar.
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.
In the Hebrew calendar, the month of Av is followed by the month of Elul.
1 week = 1 week on the Hebrew calendar. (The Hebrew week is the same length as the week on the Western calendar)
The year is unknown, but traditionally, it's believed to have occurred around 1312 BCE which is equivalent to 2460 on the Hebrew calendar. (In actuality, although the Hebrew calendar was in use, the practice of numbering the year had not yet been introduced).
There isn't any festival celebrated along with Christmas as per the Hebrew calendar, since Christmas is not tied to the Hebrew calendar. (So every year, Christmas falls on a different date on the Hebrew calendar).
A Hebrew calendar website can help you with that.
In the Hebrew calendar, the month of Nissan immediately precedes Iyar.
The Hebrew month Adar is the same in all countries. It is a Spring month.
In Israel, both the Gregorian and the Hebrew (Jewish) calendars are officially used.