Yes, you can cut any part, let it 1-2 weeks to callus over and plant it. And one note from me - your plants look terribly etiolated. If possible, you should find them place with more light. Slowly, no drastic changes in light intensity. Start with moving them to a place with filtered sunlight. If they will be ok, after 2 or more weeks try to give them few hours of morning or afternoon sun. When properly acclimated, they should take many hours of sunlight without any problems and will look much better. :)

Looks like red spider mite damage. With that much damage, you should be able to see the nasties creeping around with a magnifying glass. I used to be able to see them with my bare eyes, but that is years ago ;-). If you find them, put some rubbing alcohol or old fashioned Listerine in a spray bottle, and wet down the whole plant. Keep it out of the sun for a day or so. Repeat after a week, re-repeat after two weeks. Hopefully that will be the end of the trouble. If you don't see spider mites, I have no idea what it could be.

Like when you walk out of a dark room and into the sunny patio...takes time to adjust. If its a hot sunny window- keep watering the plant and it will adjust faster. Only move away in winter to keep it from freezing. Or,keep under lights.

I do not know about the "sap" - but found this: ..."It sounds like it has extra-floral nectary glands. These secrete sugary fluid that attract ants"... Or maybe it was poked and oozed some sap. It could be dripping sap if it was chewed on by bugs - your plant seems to be healthy from what I can see in photo. Succulents store surprisingly lots of moisture and if 'injured', it may drip. Sorry, I do not know what exactly is the case - wait until others comment. As for potting, they could be in same pot or on their own. I guess you should decide if you want more pots. Just be aware that opuntia grows in large clumps and pot will fill up faster than many other succulents. They look pretty good like that. JMO, but I don't think soil you used drains well enough for succulents. I do not use soil, but may that do find it more water-retaining. If my plants, I would add at least sifted perlite. Mixing them in 1:1 ratio. It also looks to me that soil is piled up too high on the pad - but I am only looking at the photo. Is there another/different succulent in same pot? - just wondering...