When braising a dish for a long period of time, and wine is part of the braising liquid, will ALL of the alcohol cook off?

NO. All the alcohol will never burn off. Almost all is not good enough for a recovering alcoholic. Yes, you can just use more stock. If you are looking for some acidity, though (which wine adds) you might try some citrus juice, or vinegar. Many wineries nowadays are also making a non alcoholic product called verjus. If you can find some if that, it's a great sub.

It's myth that ALL the alcohol will be cooked off in any dish. There will always be alcohol left that is somewhat bonded to the water in the dish. Short of drying out your braise, you just can't remove it. I know that for braises, a significant amount of alcohol does remain. And non-alcoholic wines do have traces of alcohol in them. If you cannot consume any alcohol, leave it out and use more stock. There is a non-alcoholic wine powder, called Dry Gourmet, which does not have any alcohol, but it will not be launch commercially until this winter (?).

Good question, often asked here and often incorrectly answered. Even after 2.5 hrs of braising, 5% of alcohol remains - tiny amounts when divided by # of portions, but some people really cannot have ANY due to medications/health issues. This chart is useful: http://whatscookingamerica.net/Q-A/AlcoholCooking.htm