They will ripen better at room temperature, and then you can store them for about three days in the fridge, they don't keep very long no matter what you do to them. They are like bananas in that they ripen really fast anyway, they'll go from green and hard to black and mushy in just a few days, and they're best before they get too soft. By the time they're black, it's too late.

Yes, you can plant the seeds, and there will be variations in the fruit of the seedlings, but most of them will be similar to the parent. Some might even be better. You need to plant them right away, do not dry them out as they may die, and they need a winter's cold before germination anyway.

A lot of nurseries now sell paw paw trees -- Raintree Nursery, One Green World, Burnt Ridge Nursery, Hidden Creek, Nolin River Nut Nursery are all good choices, IMO.