It's normal for cut potato to turn dark. Cooks often put potatoes in water (sometimes with other added) to help prevent discoloration. Just like apples turn dark quickly after cutting.

If you mean they seem to be growing dark stuff on them or something like that, that might be more of a problem.

And you say they had eyes- had? Do they not have eyes anymore, as in those spots went funky? That would be a problem. But if they are nubbing up, you are probably OK.

I hope you mean a tray with water and not just a tray :) As in trying to sprout them in the sun. I know some folks do that, but you can also plunk them into some soil if you want- some folks do that too. If you just left cut chunks on a tray in a sunny spot for 6 days, that isn't good. Curing is only a couple days out of the sun.

I have to admit that I've never heard of this before. I cut my seed potatoes with eyes, let them dry for a day, dust them with an antifungal, and stick in the ground.

As for the cut sides getting black... yes, as beesneeds described, time will do that.

This is due to oxidation created by direct incidence of Sun rays. it is normal. You can use your seed potatoes.

I cut my seed potatoes and let them fall into the hole or trench. Cover them with some of the soil..

I believe it is best to cut potatoes 10 seconds before they are sown.