Paw paws don't transplant well. they do not like their roots disturbed. try using large coir or peat pots so you can plant the whole thing. Do not drown the seeds while germinating them. just damp. not drippy. like you would wring out a dishcloth to wipe the table with... that kind of damp. you need two trees for pollination Good luck. I have one tree. the other got ripped out for a septic installation and I have never gotten a paw paw off of it.
as per KYSU bulletin:
Seed Propagation
Pawpaw seed is slow to germinate, but it is not difficult to grow seedlings if certain procedures are followed. Do not allow the seed to freeze or dry out, because this can destroy the immature, dormant embryo. If seeds are dried for 3 days at room temperature, the germination percentage can drop to less than 20%. To break dormancy, the seed must receive a period of cold, moist stratification for 70-100 days. This may be accomplished by sowing the seed late in the fall and letting it overwinter; the seed will germinate the following year in late July to late August. Another way is to stratify the seed in the refrigerator (32o- 40o F/0o- 4o C). In this case the cleaned seed should be stored in a plastic ziplock bag with a little moist sphagnum moss to keep the seed moist and suppress fungal and bacterial growth. After stratification the seed should be sown 1 inch (2.5 cm) deep in a well-aerated soil mix, pH 5.5-7, with an optimum temperature of 75o- 85oF (24o- 29o C). Use tall containers, such as tree pots (ht. 14"-18"/35-45 cm) or root trainers (ht. 10"/25 cm), to accommodate the long taproot. The seed will normally germinate in 2-3 weeks, and the shoot will emerge in about 2 months. Germination is hypogeal: the shoot emerges without any cotyledons. For the first two years, growth is slow as the root system establishes itself, but thereafter it accelerates. Trees normally begin to bear fruit when the saplings reach 6 feet, which usually requires five to eight years.
the whole page if you are interested.
http://www.pawpaw.kysu.edu/pawpaw/ppg.htm