Big 'ol bumpity bump. Learned a lot after trying various techniques since I started this thread (Jesus, july?), which I'll detail here as a now-lowbie botanist.
1) Those stupid egg cartons are too small. Gotta go bigger, using small (~1 pint) plastic planter cups now which conveniently fit 3 to a egg carton lid.
2) Anti-fungals are critical. Using anti-fungal spray has almost completely halted my loss of sprouts due to them damping off. I spray a few sprits on them every 4-5 days.
3) Second critical step was dirt. Potting soil doesn't cut it, it needs something more dense that holds water better. Switched to a blend of ~20% mushroom compost, ~20% (clean) sand, ~55% potting soil, and 1-5% wood ash. Theory on the ash being that their natural environment has periodic fires, so some carbon ash should exist within the soil naturally.
4) Lots of water. See above why anti-fungals are critical. Lots of my seeds failed to progress past initial ~1mm sprout due to lack of water. I also spritz them with water once a day/every other day, as their natural environment is very foggy.
5) Lots of light. I think the sun-through-the-window wasn't cutting it, got a UV lamp mounted on my goofy bookshelf plant stand now.
6) 6 weeks in the cold. Not 5, not 4, not 4-6, 6 weeks. I get double the sprouts letting them go to 6 weeks over 5.
7) Switched from goofy paper towels and coffee filters to pearlite, and improved my handling of the seeds to reduce the introduction of bacteria and fungal agents into the fridge-stratifying and germinating seeds. This has been one of the biggest improvements to my sprout rate, those paper products are just gross.
8) Sprout goes DOWN when putting in a pot, if it gets confused and grows up, it'll die.
9) About 30% of my current seeds are sprouting (probably get up to the cited 40% with further improved handling techniques... maybe surgical gloves?). About a quarter of those actually sprout with their seed still attached, rather than sloughing it off in the dirt. I've improved this somewhat by buring them a little deeper and padding down the dirt above them a little better, but some still come up with it. I've learned that if I let them sit there for longer than I feel they should, the cotyledons (first tiny leaves) actually split correctly, at which point all I have to do is cut the seed cap in half and usually it'll pull off after that. So basically, let them sit and see what happens!
10) Don't be upset if sprouts die. That's why you get 10 thousand seeds. Just keep planting.
11) Don't be afraid to do one sprout per pot. I was getting a lot more die-offs early on so I put several in a single pot, and damn near every one sprouted resulting in me having to manually separate tiny amounts of dirt with a sprout out of a single pot.
I now do pearlite with a little water in a baggie, let the water sit for a minute to kinda soak into the bag, then drain of most of the excess (leave a little in so there's still moisture as the seeds absorb it). Seeds need to not be sitting in water while stratifying or germinating or they'll drown, but they do need to be touching wet stuff to absorb properly. Germination happens anywhere from a day to a month after pulling from the fridge. Any seeds remaining after a month/month and a half get dumped outside in a corner of my garden and covered lightly in the event I get a random late sprout.
Pics of current sprouts, a few with actual leaves beyond the cotyledons. The largest two are about a month old. ~1mm sprout to actually poking up takes about a week, another week to stabilize the cotyledons, and another week or two to actually start sprouting those little leaves.