I don't know which cacti they are, but I just use gardening gloves. Yes sticking-out spines will go through them. The spines on a small barrel cactus or mammary cactus where they are wrapped around more, are less likely to go through gloves.
Usually I'm working with opuntia (see this question for photos), where the spines stick out more, but there are usually gaps between the spines where a pad can be held without being spiked. Opuntia are also pretty hardy when it comes to this kind of thing. They get a bit of a rough treatment if I'm cutting pads off for cuttings (depending on your cactus type, this might be something to consider?). Also the "cinnamon bunny ears" opuntia requires sacrificial garden gloves. It doesn't have spines but has nasty hairs that stick in the gloves. Often it is best to just throw the gloves out after use.
Newspaper does sound a good idea - especially if it is like a barrel or mammary cactus with the spines wrapped around. The paper should deflect them - the opuntia spines may still go straight through the paper.
So yes, I would try to loosen the soil and carefully pull the cactus out (wrapped in newspaper). Do the whole thing over newspaper to catch the soil and debris.
Edit after seeing photos:
Yes the cactus looks like one where the newspaper trick would work. It is more of an upright mammary cactus to what I have (or see growing locally), but it has that wrapped-around "bird nest" pattern with the spines.
I see there's some finger space around the base, so I think you should be able to turn it upside down and tap the root ball out of the pot.
You may be able to split the side growths off once it is out of the pot. A knife would help make a cleaner cut. (my opuntia don't seem to mind but as I say, they're pretty tolerant of a lot!)
I looked in "Cactus" by Elisabeth Manke, Barron's. This says M. gracilis grows to about 4 inches. No specific propagation instructions, but says Mammillaria can be propagated by seed, grafting, or cuttings. I'm not a fan of cactus grafting - visually it never looks right.
If the jade cuttings took so well (sounds very much like opuntia in this respect), then I wouldn't be afraid of bits breaking off. If they do, you have more cuttings. Soon you'll be giving them away to your friends. Perhaps put fewer in each pot next time. Perhaps 3-4?
(The optuntia pads are obviously a lot larger, but I plant those one pad per pot and get >90% success rate. )