Syn,
Having a cricket bin with the top fully open, or with a screen lid, is the way to go. I mention that our bins are plywood (and large, usually 6 ft long). The top edge is 4" of acrylic, kept clean, and the crickets cannot cross it. A new sweater box will also be unclimbable, if at least the top 3-6" are kept wiped clean. A little bit of dirt, water spots, heavy dust, worm chow dust stuck to a wet side, etc., and small crickets can climb it. Larger crickets need more residue, as their own weight works against them.
You really do not need to clean out a cricket bin unless it gets wet. Otherwise, just pour in more of the worm chow. I say use Purina worm chow, as Purina stopped making cricket chow in 50lb bags a few years ago. Other local grain and feed providers may have cricket chow available inexpensively through your local feed store, but some of it is pretty poor, as the crickets won't eat 25% of it. Other times, I have been able to get excellent regional product. But Purina worm chow is a safe product, well eaten.
We glue our cardboard crates as honeycomb. You may want to scrape out the used bottom of a cricket bin every 15-25 days, or just let it pile up, lifting the honeycomb every week or so, as it will be 1/2" sunk into the dry debris. Dry cricket waste is really not a problem though. When its wet, as in a leaky water bottle, its a huge problem. Clean the wet out, get a non-leaky bottle, and get back to business. Change those water sources every 2 days !!!!!!
When kept at 85-88 F, crickets mature to laying their own eggs in 42-45 days from when they hatch, with a total life cycle of about 55 days. Once they "wing out", as in grown wings, they can usually lay (and males chirp). Cooler temps extend that up to 100 days or more, simply by reducing the growth rate. But then you need a larger cricket colony, as everything takes longer to get to the size you want.
Screen tops may help keep some crickets in. But often, if you raise a lot of crickets, its to keep adult dermestid beetles out. Dermestid larvae kill young crickets, by the 1000's. Also known as cadaver beetles, they eat other living (or almost living) things. Taxidermists use them to clean skulls, etc. They can ruin a cricket bin of smaller crickets in 1-2 weeks or so, if allowed to propogate in quantity. You never get rid of them all, but you can minimize them with well-fitting screen lids.
It takes a little practice, getting your set-up right, learning from mistakes, etc. The article linked was pretty good too, but was wrong on needing high protein to raise the crickets. Not so. Just have a good gut-load (we call it the "Last Supper" here) for those crickets to be fed to chams that day. I go thru 10K crikets daily (I don't really count them, and its nothing to blow thru 5000 crickets of 1/8" or less) and all my advice is based on experience here. A good cricket set-up for the hobbyist might be 6-8 sweater boxes on a set of shelves, kept warm, and hopefully not PO'ing the spouse. Good luck.