Could you post a picture of the "moon cactus" that you have? Sometimes they're grafted onto something other than a hylocereus undatus. As for the top (gymnocalycium), if it's brightly coloured then their isn't any point in trying to replant - it can't produce chlorophyll, so it will die. If it's darkly coloured or has some green to it, then you could try to root it by nestling it into some perlite or your choice of rooting medium, keeping it in a bright location with indirect sun, then lightly misting (NOT watering) the rooting medium PRN until roots are present. As for the hylocereus - it should be quite happy after being freed from the burden of the gymno. You can sprinkle the cut area with cinnamon as an antibacterial measure. If you're growing indoors, direct light is best. Water when the soil is quite dry. It's pretty good at letting you know when it needs water; it will start to feel a bit softer and get slightly wrinked when it needs a drink. What sort of soil do you have it in currently?

This one looks like an older stem cutting that may have been rubbing on the ground or something or maybe some minor freeze damage; I don't think it's anything to worry about.

It's not easy to id which Pitaya variety will bear edible fruit. All cactus species will flower and bear fruit at some point in their lifetime, same with Pitaya. However, some Pitaya do not produce edible fruit (i.e. too small, too thorny, etc.) but they all should send out these spectacular flowers which bloom late at night. Like all cactus species, Pitaya thrives in dry, hot and poor soil condition. If there is too much water or fertilizer, they just get massive (like the one in your customer's backyard)and climb higher and higher without producing any flowers or fruits. Here in Hawaii, in dry locations on the island, local farmers grow Pitaya dragon fruits in cinder gravel and let them climb on lava rock walls. Sometimes hand pollination of the flower is required to make the plant bear fruit. I have a red, yellow, and white variety growing very well in my garden (yes, in plain cinder soil) and they had produced edible fruits. You can just cut a mature piece of Pitaya about 1 foot length, stick it in a pot filled with sandy soil, water sparingly and it will root within 2-3 weeks. Or you can find a larger piece on your friend's plant that already has clump of aerial roots. Pitaya love send out roots to attach themselves onto rocks or tree branches or whatever they're crawling over! Good luck. Aloha!

If it came away like that it seems the rot was in there. Normally those parts don't separate so easily. But yes, you can leave them drying off for quite a while without any detriment to the plant (in shade). And if you lay them on damp soil roots will soon start and go down.