First of all, shredded cardboard is rather dry so you have to make sure that you are not starting with too little moisture. Also cardboard doesn't have much bacteria to start decomposition and preprocess the material for the worms (worms only eat material that's already decomposing), so I'd add a little bit of dirt and a few greens at the start as well.
Second, I think it's better to start with less initial bedding material. Processing a lot of carbon in one go is more difficult than spreading it out over time, because you want to keep a C:N ratio of 25 and a steady flow of decomposing material. Also lots of wet cardboard may clog up your draining system.
How much material worms can process depend on various factors:
- amount of surface available (red wigglers like to stay just below the surface)
- amount of worms (this will regulate itself over time if conditions are right)
- preferred temperature for the worms (for red wigglers this is 55–77 degrees Fahrenheit (13–25 degrees Celsius)).
- preferred moisture level (about 60-70% is ideal)
- ph level / type of food (don't add too much citrus fruits at a time)
- amount of air (make sure the contents don't turn into a soggy lump)
Red wiggler worms can eat about half of their weight in food every day, but there has to be already decomposing material available, so you need to feed regularly.
I've read that in an up-and-running bin there generally are around 1.5-2 lbs of worms per square feet (7-10 kg per m2), so with 2 x 2 = 4 square feet and 5 lbs of worms you are already approaching maximum processing speed. This means that you'll be running at maximum processing speed in a few weeks, provided the conditions in your bin are right.
How much bedding/compost you pile up over time depends on what your goal is. Personally I'd remove already finished compost once or twice a year (or more regularly in case of a flow through bin) and keep the bin between 1/4 and 3/4 full. There is little point to filling it to the top.