What you had was a mutant Gymnocalycium mihanovichii, cultivar 'Hibotan'. It has no green chlorophyll, so other pigments are visible, causing the red color. It has no way to feed itself, so it is grafted onto rooted stock, typically a Hylocereus. The 'ruby ball' cannot survive on its own. Since it has rotted, all you are left with is the relatively uninteresting root stock. Gymnocalycium mihanovichii cv. 'Hibotan'

You could graft the wee ones onto the green stock plant or onto a new stock plant. Leslie is quite right regarding the fact that they cannot survive on their own (they lack chlorophyll and thus are incapable of producing enough food via photosynthesis).

Hi: It's a perfectly normal development, it's called an aerial root. Thoughts vary on why/how these occur. Some folks feel it helps anchor to something while the plant climbs up the canopy (these are tropical or jungle dwelling cacti). Others feel it helps the plant absorb extra moisture in the air. Either way, it's normal, healthy, not a problem or anything to worry about. The new growth looks good from here! Lena

The red top is grafted plant, totally different from the green one. Red doesn't have any chlorophyll and can't produce food, so it is grafted on different rootstock to help it survive. These grafted plants do not last many years. Top plant often dies after few years or so. If it got rot infection, it may be done for now. Rot could be usually cut off, but not sure how you would do it on this plants as it seems to be quite close to base of red cacti. You may try to hollow up rotted parts, see how much and how deep into plant it rotted. If whole red cacti is soft/squishy, it may be rotting thruout. You could cut it all off the green one, and that green plant will continue growing (unless rot got into it) - the rootstock is much hardier plant.