Whether or not 3 nights in Carcassonne is too much depends on what you're interested in, and what you plan to do when you're there. And, I guess, how you will get around.
I have been to Carcassonne twice, and about 1/2 day (or less) was enough for me. I like older castles and learning about history; I'm not much for tourist tat. I am very interested in the history of the Cathars and that region, but the town of Carcassonne was heavily and somewhat inaccurately restored by Violet le Duc in the mid 1800s. So, if you care about that kind of thing, the old town is impressive but doesn't look the way it originally did. As mentioned, there are many "Cathar" castles in the area to explore, but you would need to take a tour or a private guide, you couldn't get there by public transportation (and while interesting and scenic and well worth visiting, many do not actually date from the Cathar period). Check out Queribus and Peyrepertuse to start, and closer to Carcassonne, Lastours.
Tours for 3 nights is ok, again depending on what you're interested in. I prefer Amboise as a base for the Loire and for visiting the chateaux, but I think public transportation would be an issue there, too, again unless you took a tour or got a private guide or driver. Amboise is a gorgeous town, and would be a nice variety from the larger cities that you have on your itinerary.
I've only visited Bordeaux to leave from for visiting the Bordeaux wine country and the Dordogne, but I have heard that it's an appealing city. Though also, if it were me, I would rent a car and explore the wine country. (And if it were me, I would skip Colmar and Lyons and head straight to the Dordogne area, which I love for its prehistoric and medieval history, beautiful scenery and delicious food, but an area that would be difficult without renting a car.)
I envy you the amount of time you will be able to spend in France - enjoy!