1

Euhm, have you ever been in France in winter? You will most probably have snow. So you are talking about mountainbike trails? Well, if you are prepared to cycle in minus 4C, in snow, or even worse, rain, and then sleep in a tent, yes, you can do that. Just doesn't feel like much fun for me :)
And not sure if campsites are open in winter.
I would go in april/ may, nicer weather.

2

Cycling in the departments of Aude and Ariege (where you'll find the chateaux) are hosts to the Tour de France most years, one route or anothe. Cycling on the main routes should not be a problem, but snow might be.
I've stayed in the village of Montsegur in January and they often have enough snow to close the roads until the snow plows can get out.
As far as camping, most camping sites would be closed (a lot of businesses as well)
There's a hostel of sorts in Montsegur but it's closed from November to April...as I think many places (aside from the hostel in the Cite in Carcassonne).

Not one of the 5, but interesting and quite accessible, the chateau at Roquefixade isn't that far (as a crow flies) from Montsegur. There's a gite d'etape that's open all year and the late mayor of the town was quite an authority on the Cathars.
http://www.roquefixade.fr/index.htm

Is your main interest in the castles or in the Cathars? Do you speak French? (if yes to the second one, I could provide info on people in the area to speak with).
Anyway, hope this gets you started.
ps...there's a forum "On Your Bike" you might post there too...
The chances for rain in the area are fairly high that time of year so you'd need to have wet weather gear and very visible lights. There's train service from Carcassonne down to Quillan (do NOT even think about going to Chateau d'Usson (not one of the 5) as the highway that follows the Aude river up to the chateau is rather narrow and often on roads cut out of the rock...not safe, esp when it's a route used by those going to the ski slopes further up in the Pyrenees.

Here's a good beginning site for Ariege:
http://www.ariege.com/pyrenees.html

And for Aude:
http://www.audetourisme.com/en/index.php

A chatty newsletter for Aude is http://www.audeflyer.com
It will have listings of what's on and if you look back through old issues you'll get a sense of the weather.

I have links for some English language blogs in the areas. You don't say where you're from but your English is excellent so I'm thinking English languages sites wouldn't be a problem for you. If you'd like the links, send a PM.

If you're not already familiar with Michelin maps, then forget Google and visit:
http://www.viamichelin.com
You can get an idea of the terrain around all the castles (there's a reason they were able to defend them for so long...) and find other info. Michelin does give you the option of cycle routes, I think (scroll down the options on the left side).
There have been a number of inquiries about this area. You might try using the search function, but use the keyword Languedoc..you'll probably find links listed which give more info on weather, recommendations, etc.