Help is at hand - I keep a whacking great old coolbox full of lobster roaches.
Yes, they do breed like crazy, about 25 or so a litter, livebirth. They need about three months to reach sexual maturity, and to get a good colony going you actually need two boxes to begin with - one for your animals to eat, one to start a colony. Stops you from dipping into the colony too soon. It's not really easy to sex the roaches, so the twobox technique helps. Don't worry if you see what looks like long yellow/orange poo hanging out - that's an egg cluster from a pregnant female just cooling off. Bit like beardie gaping.
Yes, they are climbers and they're pretty quick - winged adults can also flap quite a distance, but mostly downwards. Very simple to keep them under control - use a heavy layer of vaseline all around the top 2 inches of the box, and also around the wire of your heating mat. The warmer you keep them, the faster they grow. Under 10C, they die, but try not to get them over 110F. 95+ and you'll see the breeding go insane. They need a lot of fresh fruit and veg for water, rather than watersupplements, and cheap pond fish food sticks are great for bulking out the diet, as is cat or dog biscuits. I personally chuck all my kitchen ( Not meat or onion/garlic) food waste in the tub.
Hope that helps, any questions, just ask.