I'll take a crack at the questions for veterans here...
Here's a
link to my trip journal from my last year's north to south trip. It'll have some more details on this stuff.
We did a rental car to get to El Paso, but Amtrak also goes there. That's the closest city with any kind of connections to the outside world. Depending on how serious you are about the official route you can cut up toward Silver City and miss some desert highway.
Snow just depends on the year. It was almost 100 percent cleared out for me last year starting around the same time--just had to hike over a couple of drifts. If it's still significant when you start, you may just want to do short days in NM and by the time you get to the big mountains you should be OK. 50 miles is a long way to go at first so your body will thank you for the short days anyway. The route doesn't follow the top of things for long usually so even if snow is significant, worst-case scenario should be that you'll have to hike-a-bike a few miles up top.
About bears -- I always just did the triangle thing where you camp in one spot, cook in another, and put the food in a third, with the food and cooking spots downwind from the camp. I just put all the food in a garbage bag on the ground. I never had anything get into it, but I may just have been lucky. The bears definitely are out there and with a bigger group you may be giving them more to smell than we were. Also I'm not sure all campers would agree with my system--it should keep the bears away from you but still can cause trouble for a bear that finds out it likes people food.
Good luck... it'll be a great trip. You should run into all the racers too, which is cool. Are you going to be posting any updates?