Just put the papaya seeds in some dirt and keep them damp, but not wet and soggy. Also, the fresher the seeds, the better the germination rate. The papaya I bought at the grocery store got pretty ripe before I opened it up for the seeds, and some of them were already germinating while still in the fruit! Hope this helps.
Besides heat, the biggest thing you need for germinating pepper seeds is patience. Germination can easily take 3-6 weeks depending on the species. I've tried the damp paper towel method... It didn't work for me. Just a seed starting mat, water, soil, and time works best for me.
Although I know very little about papaya, this I do know: the trees must be in pairs to pollinate. If you got seeds to germinate, how would you know which trees (male or female) you had together? And, once they are adult trees and producing, they are a lot of work. The fruit is extremely fragile; they must be wrapped, while on the tree, to protect them from worms and bugs. This is probably why the fruit is so expensive to buy.
As for the pepper seeds, that's easy. I bought a grow kit - one with the compressed peat pots in a molded bottom and clear, plastic topper. Just follow those directions. Mine took about 2 weeks before germinating. I now have them separated in larger pots. I used seed starting mix - probably not the best medium but what I have. By day, they go in a window sill. At night, I put them on styrofoam trays (keeps water and soil contained) in a warmer spot.
You do know that you can freeze the peppers, without any blanching? Just wash, remove seeds, and slice or dice and put in freezer bags. Marvelous for soups and stews in the winter, and stir-fry any time.
Gotta chime in here a bit since I'm growing and harvesting papaya right now. I got the seeds from a store bought fruit and tried my luck. Out of 10 seedlings only one survived and grew to maturity, which is not rare with papaya.
While it's true that you need a female and a male tree to bear fruit, there's also a hermaphroditic tree, which is self fertile. That is mostly the hawaiian solo papaya, not the mexican large fruited one. You can see when the tree flowers what sex they have because the flowers have different shapes.
You can protect the fruit with a net, so the can't be attacked by insects, but me, having had not the time to do this and here I am eating my own papaya.
The climate here in South FL is great for growing papaya.
And Julie, I use seeds from the store bought papaya and put them in seed starter mix soil or other soil and they will sprout with a germination rate of 90%. If the seeds are older you'll have to wait longer and have way less germination rate.
Good luck with your seeds and don't give up if the first try doesn't work out, try again, you'll succeed!