Although we cannot be 100% sure, as no cameras existed in that time, I would guess it's not probable. Because the Assassins depended on the ability to "become one with the city's crowds," such a bold and uniquely designed outfit would raise unwarranted suspicion. They may have worn similar garments within Alamut's confines, but there's really no way of knowing whether they did or not. The outfit is a creation of Ubisoft to glamorize the Assassins.
No, it's a mechanism thought up by Ubisoft. They did, however, use another weapon from the game frequently (you can read about it in the Assassins page), the poison blade. Not to mention the bracer containing the blade itself is quite noticeable, and could have drawn hindering attention it itself and the Assassin.
No, the symbol was made up entirely by Ubisoft. However, the Assassins did use a different symbol, the Hajal bird.
It's entirely possible, in fact there are many in existence today. In this age of technology it would be much easier to round up a group who are willing to form a Brotherhood. And now that we have cell phones, email, and other forms of instant messaging, contacting other Assassins in distant lands would be more convenient for transferring information. Weaponry has also advanced, which would give an Assassin a variety of assassination tools.