Update:
After all of the seeds I had planted directly in the ground last year failed, one of the backups managed to germinate in a bag and was planted in the ground late last season. It made it through the winter with two tiny leaves and has recently added more. (The plastic soda bottle is just a sleeve to protect it from critters--there's no bottom on it.)
In late winter I was sent a bunch of 'Mango' paw paw seeds which I decided to germinate indoors this spring. I put them in a plastic container (indoors) on moist compost, and soon saw mold forming on one seed, as well as on the soil. I removed everything, wiped the seeds and the container with rubbing alcohol, and tried again on moist Perlite. That did the trick--no mold, and 100% germination looks probable.
To accommodate the long tap root, I planted a couple of the germinated seeds in double-decker pots made from soft drink bottles (one seedling just broke the surface a few days ago); others I've planted in quart milk jugs and whatever else is handy. Several have been planted directly in the ground, but all will have shade, whether from a cover or nearby tree. I'll update in a few months to report on how they handle Tucson's summer heat.
Mouse-over images for description.