The MAJOR parades, unless things have changed, ALL follow the same route: down St. Charles to Canal and then along part OF Canal. They don't go into the French Quarter at all due to fire and other regulations/precautions.

The several years I lived in the city I saw a variety of "activities" including OPEN screwing on a couple of the side streets but that was the "worst" of things. I would not hesitate to have children watching the parade(s) themselves. Drunkenness is the major theme for many folks, especially within the French Quarter itself and there have been more than a couple people who have fallen off balconies along Bourbon Street.

Go to Google maps and check out these areas. Then drill down close and you'll see restaurants. Mouse over the restaurants and you'll likely get ratings and other feedback.

And no, considering what I've done or attempted in the Quarter, I wouldn't take the kids anywhere near Bourbon Street (closer to the river and Jax Square should be ok).

The quarter during Mardi Gras will be much more than the usual drunkenness you run into every night there. You will see blatant drug use, young women on balconies encouraged to remove their tops and bras (which they will do) - and perhaps more. Petty crime is rife - and muggings if you go onto the side streets. Fights and other random violence is rife in the streets.

I would not go there myself at all - never mind with teens. If you do go leave any valuables in your hotel room and women shouldn't carry purses at all. (I saw part of an episode of Cops n NO during MG and it was the worst thing I have ever seen - not a fun street party event - but sordid drunken violence.)

I would really rethink this unless the teens are sophisticated, city wise, mature and large enough to defend themselves/not look like victims.

I was in Thibodaux for Mardi Gras in 2005. It's much smaller, and not even the same thing. But it was all about family fun. I stayed with the Naquins:

http://www.naquinsbb.com/

They know everybody in town and took me to the warehouse where the floats were being prepared. It was fabulous.

Hi-
You're right about watching (most) parades on St. Charles Avenue. It's much more family-friendly and you will not have to worry about women flashing (and some men!) But Endymion rolls on a different route- the non-tourist spots for that parade are in the Mid-City areas.
However, you don't really need to buy tickets to watch the parades. The crowds are part of the fun! If you really want reservations, try Michaul's, Zea's and I think some hotels like the Intercontinental offer package deals-- food, viewing spots and most importantly, bathrooms! There are also viewing stands in front of Gallier Hall and in Lee Circle which are free and open to the public.
And to keep track of the parades (and to stay busy during your wait) download the WDSU Parade Tracker app. Yes, I work there but it's a GREAT resource that will give you real-time updates as the floats progress. Plus it's free! (WDSU.com also has maps of every parade for your planning purposes.)
I might reconsider staying near the Quarter since you'll be with teenagers. During the morning the rowdy visitors will be sleeping in but after that it will be VERY crowded and no fun, not to mention very bawdy. There is a reason our tv cameras keep a wide angle shot of Bourbon street- no close-ups allowed!
You could stay Uptown and catch daytime parades on Saturday, and parades all day and night on Sunday. You'll also deal with smaller crowds on Saturday Uptown since most of the city turns out for Endymion that night. If you stay near the Quarter just know it will be tough to get to other neighborhoods due to parade traffic and crowds.
Your restaurant choices are great and enjoy the WWII Museum- you will spend more time there than you think!
One more tip- don't walk around the Quarter with your smart phone out. There has been a spike in thefts since most people are paying more attention to their phones than their surroundings.
Have a great trip!