It was welcoming to read a well-written post and your desire to come to Qatar. Let me try to answer your queries one by one:
* The rules with regards to age of getting hold of a driving license are not flexible and as such you will simply need to wait it out till you turn 18. I know of some Qatari young people in your age bracket who are unable to get hold of the driving license due to this age requirement.
* There is no such thing as a "teen" culture existing in Qatar. One thing you will realize when you get here is that the majority of people living in Qatar are foreign nationals. As such, mingling of teens from several countries and having a "culture" of their own, is not visible. The local teens may have one of their own, but I am not aware of this.
* Yes, you can learn Arabic here is Qatar. You will not be able to learn it from your interactions with people as most people speak just English and many locals also do. There are private institutions that teach Arabic. You will face some difficulties initially in picking up the language for several reasons. First is the script which is written from right to left. Second is the phonetics where many Arabic words are pronounced through sounds deep in the throat.
* You will not be allowed to work in Qatar. Your age will not allow you to find employment in the country. Secondly, one needs a Work Visa. This can be provided only by a sponsor. There is no such thing as "teens" working at jobs.
* Yes, Qatar has have a dress code. You cannot run around the city or on the beach scantily clothed such as in bikinis or see-throughs. One must dress modestly.
* Among the many don'ts: You cannot drink in public, walk or drive around under the influence of alcohol. Doing so, may get you arrested, fined, jailed or even deported. You will not be able to find alcoholic drinks just off-the-shelf at the supermarkets or at other stores. The show of affection in public is an absolute no, no. You will not get to see fist-fights taking place. You can walk safely down any dark lane past midnight all alone without the slightest fear in your mind. No bag-snatching takes place. Qatar is country which has a very low crime rate.
In the end, Qatar is a small, very peaceful country. The country stretches just about 160 kms (100 miles) from north to south and 80 kms (50 miles) from east to west. You will love the environment and when you go back home, you will be taking along with you lovely memories of this place.