Mine turned quite red/brown when I first transplanted them into full sun. They had been growing in full sun in the pots, and now are PIG. Now they have adjusted and are quite green again. I don't know if it was the spot I moved them to or a little shock from the transplant, but I was pleasantly surprised they returned to green. I know you recently transplanted, so maybe just give them time and see what happens with the color, especially since they are otherwise healthy. Mine were brown for about 6 weeks.

Yes to above recommendations. If you take it out of pot before watering, get a bowl with tepid water and swish the root ball with hardened soil in it. You could leave it soaking for a while if necessary, it won't kill the plant. Soil should soften enough to be able to pick off. You can help yourself with chopsticks or wooden/bamboo skewer. Air dry as above after getting as much original soil off as possible. Do not water for a while after repotting - there will be enough moisture from washing the soil off, and it also will give plants roots chance to heal - some will break while handling. The purple/red color should be no problem, yellowing leaves usually are waterlogged. Putting it into grittier mix should help with that. Filtered afternoon sun is probably not enough light, plant is starting to etiolate - stretch. After repotting, put it slowly into more and more light. Succulents are (almost all) sun lovers, more light is better. The 'little bud' is likely a new branch, leave it to grow more and it could e removed and potted separately later - or leave as is, either way is OK.

What is the mix the plants potted in? It looks like it holds water. I would have it in something extra gritty like a 60/40 mix of perlite/pumice and cactus&succulent soil. How often are you watering? These go dormant during summer so I wouldn't water too often. Does the pot have a drainage hole? These like a fast draining soil and proper ventilation. So hopefully there is a drainage hole. Where do you live? What kind of light is it getting?

Yes, cut it off ASAP. Rot can spread fast. Cut just above rotting spot (red line), and keep slicing off thin slices until flesh is clear of any dark spots or lines. Make sure there aren't any left, or it may continue rotting. Use sharp blade and wipe off with rubbing alcohol between cuts. If flesh is clear, let it callus/heal/scab over without potting. If worry, you can dust lightly with sulfur or ground cinnamon - both are fungicides. This is not absolutely necessary, but may help. When calllused, stick it into well draining mix. It should contain very little soil or other organic matter. If you don't have anything else, mix Cactus and Succulent soil with sifted perlite in 1:1 ratio. Cuttings without roots should not be watered until they start growing roots - by then, there is usually some new growth showing. You should check these threads -BEHEADING PLANTS 1&2- to see many photos and get additional tips.