Sometimes if you plant one on its own, it'll lose its color, start producing chlorophyll, and live. Other times, it won't. Generally, purple ones should be okay (because I think that's their normal color anyway?) Red, not so much.

You CAN regraft them. I don't really know how, though, so someone else will have to explain the process.

The offwhite base on the green cactus it's grafted too is probably corking. If it's mushy, it's rot. If it's still hard, it's corking. Corking isn't really dangerous although it's possibly indicative of a more chronic problem such as overwatering, or sunburn.

I don't grow these chlorophyll-free plants, some say that they are short-lived but I don't know if that applies to all of them or if it is the combination of this Gymnocalycium species with the Hylocereus stock that has a limited lifespan. I once degrafted a dark purple one, and it looked happy, but I would not try that with a completely red plant. So, the "broken off balls" (offsets) should be grafted, not rooted. I hope they are still attached to the mother plant, they need to be in good condition for the graft to succeed.

Anyway, if I would make my own grafts of a "ruby ball" on a tough stock I would try Opuntia humifusa (or the variant O. compressa). That one is probably the hardiest of all cacti, and it does not seem to induce as much offsetting as some other grafting stocks. Echinopsis/Trichocereus species are often used too, they are relatively hardy. Three less hardy alternatives are Selenicereus, Harrisia and Myrtillocactus geometrizans. I have only tried various Opuntia, Selenicereus grandiflorus, Pereskiopsis spathulata and a Hylocereus (the two latter is not useful in this case).

There are many good places that describes the process:
http://www.cactus-art.biz/technics/Grafting_on_opuntia_compressa_index.htm

http://www.cactiguide.com/forum/viewtopic.php?t=6799

http://www.youtube.com/watch?v=IPy_01OSDUg

http://www.youtube.com/watch?v=Bda1k09CqaM (in Dutch, but it shows the technique)

This one can be of interest to you ...
http://www.youtube.com/watch?v=3nQYJ7PmQEQ

Be prepared for a few failures, it takes a while before you get everything right. Now is not the best time of the year to do this, the plants need to be growing. Late spring/summer is better.