I can not see my old man either, he is the really hairy white one. It is alive and it does things like get taller and last year it made a baby on it's side ....so maybe it is a she. I can't tell under all that hair. Maybe take a measurement, or a picture of where its growth point is and see if it gets taller or the arrangement of needles changes. And put it under the porch roof for those daily tropical downpours.

I do not grow cacti, so can't speak much about specific watering frequency. But I expect 1:1:1 gritty mix indoors would be too wet for a desert cactus. Best to increase the amount of grit and decrease the bark or napa. Make sure you sift out the tiny particles. An 8" pot is quite large, but I suppose it depends how fat the cactus is. Moving up just an inch or two in pot increases volume significantly (often 50-100% more). To determine watering frequency, I recommend caring for it yourself for several weeks, if possible. For friends I often use a different/less demanding mix and it takes me some time to figure out the frequency (or reasonably closely).

In my opinion, the second one is goner. However, you can save it in a way. Get a sharp, clean knife and cut the top off above the bad area. The interior should look damp and clean with no sign of the deteriorated area of the plant. If you still see a bad spot, cut again until you see all clean tissue. Set it aside out of the sun for at least a week so the cut edge can dry up. or callus. Get cactus and succulent mix from the nursery (you can get a better mix if you also get perlite or pumice to mix with it 50/50, but at the least get commercial cactus/succulent planting mix). When the bottom of the cactus is completely dry, plant it in the mix you got. Do not water until your cactus has started to grow roots, which could be weeks. Don't worry, it has a lot of moisture stored so it will survive. If you water, it will rot because it has no roots to take up water. As for pots, you could use that little pot it was in if it has holes in the bottom and if you do not double pot it as it was. If you buy a pot, get a small clay pot. Your plant does not want to be in a large pot. It died because of being double potted and the soil isn't fast draining enough and was holding too much water.. Also, it was probably not getting enough sun. It needs full sun, but transition it gradually to that much light. Others know more than I, but I would repot the euphorbia in a slightly larger clay pot. The little buckets are cute but are not helping your plants survive.

Normally I would say wait till spring, but I think this guy needs help asap. You wait till spring you may have a shriveled cactus. How did the roots look? Ok now I'm curious if the bottom half is dry I mean dead dry. Like no more life left. If the bottom part is dead you need to turn it into a cutting. The bottom looks desiccated. I don't think the roots are working much if at all. Sometimes with cactus if not watered during their growing season and when it's hot out the roots will dry up and turn to dust. I've had this happen a couple times. If not getting water they will just shut down. It takes a long time for them to die this way, but even cactus will eventually die without water:(. If that lower third of the cactus looks and feels alive. Start giving it little sips of water about once a week. It probably feels like you should dump water on it, but the cactus can't absorb that water due to the lack of roots. So the little sips will jump start it. Keep it warm and in your brightest window. Your soil needs to be amended. Looks very water retentive. Get some perlite and or pumice. Do a mix of 60% perlite and or pumice, 40% cactus mix. Get a small terra cotta pot. The pot it's in is too big and will just complicate things. A pot that's 4 inches wide would be fine. Terra cotta will usually always have a drainage hole, and allows moisture to evaporate. Rinse and strain the perlite and pumice with an old kitchen strainer:). If the bottom is desiccated cut the bottom third off. If the flesh looks healthy not brown or dry spots stop cutting, but if the flesh is bad keep cutting up till you see healthy flesh:). Let the cutting callus over for a couple weeks in a room that's bright, dry, and warm. No direct sun till roots form. No water till roots form. They don't have roots so they can't absorb water. Once a callus has formed over the cut set on top of the mix, and wait. Roots can take awhile on cactus. Don't be alarmed if a month goes by and no roots form. It's not the growing season so it may take two or more months. Don't go checking every week they won't be there. Slows down the process when constantly lifting and checking. Plus a side note. One time I was checking a cactus cutting for roots and it finally had them. I was a little to rough while checking and they all ripped off. So that set me months back:(. All of a sudden your cactus will perk up. Then begin to water. You will know it's pretty noticeable on cactus because they will plump up when the roots start searching for water;). Goodluck:). If you have any questions just ask. Your other cactus looks like pics I've seen where cactus have red spider mite damage. I've never had it. I would keep that guy away from other plants.