I could guess at the names of your plants, but others know much better than I about identification and care, but I'll do my best in a general way.

Make sure all your plants are in a fast draining potting medium. Best is to get commercial cactus and succulent mix and combine 50/50 with perlite or pumice which should be available at garden centers. (Sift out the small particles in perlite or pumice without breathing it.) If you just want to use commercial cactus and potting mix, be extremely careful not to overwater. Clay pots would work best in that case.

I cannot tell your pot sizes, but remember if you put a small plant in a large pot with a lot of soil, it will most likely not do well. All pots should have drainage holes.

The second picture--I believe that is a haworthia but have never seen one like that. Was it painted or is that natural? I would get a small clay pot for that one and repot it. Get another small clay pot for the cactus. Pot them separately. The cactus will take full sun, but not the haworthia. If that's a painted haworthia, not sure if it can live but give it a good chance.

As for the beheading, that one can grow if you put the stem in a small pot of the soil mentioned above, in a bright but not full sun location. Do not water until you see a sign of growth or believe it has roots. Could be weeks. If you water with no roots, it will rot, not grow. The remainder of the plant could sprout again, not sure, but it won't be happy double potted like that.

Water your plants only when dry, letting the water run out the hole in the bottom of the pot. If you are not sure if a plant is dry, stick a wood skewer in for a short time, then pull out to see if it's damp. If damp, don't water. All your plants can be moved into full sun, except the purple one.

Know that these plants store water in their leaves and stems, so if they are overwatered and their roots are too wet, they rot.

If we know where you are, your gardening zone, it would be helpful too. Sorry if this is too much information.