Depending on what kind of shape you're in now, you might gain or lose some weight once you begin training, so that could help or hinder you in weight classes.

You definately will need to put the effort in. You have to remember that trainers are people too, and don't want to waste their time training someone who doesn't have their heart in it, or with someone who is going to learn this stuff, then just quit because they don't feel like it. I can almost guarantee that once the trainer sees that you're in it for the long haul, and really care about getting into boxing, theyre going to notice you, and start helping you. It could never hurt to ask some of the other boxers (preferably one of the guys who has already had some fight experience, as theyre the most likely to actually know what theyre doing) to teach you some of the basics of boxing, and some basic routines on the bags.

For running, its not so much the distance as the time you spend doing it. If you can run a mile in 6 minutes, it's not going to help you very much. I think the basic numbers to run off of are about a half hour of cardio to start burning fat. Run as far as you can in that time period. If you get too tired, or winded, slow down to a pace where you can continue. As soon as you can, kick it back up again. You'll find pretty quickly that you'll be running faster and longer the more you continue.