An ordinary refrigerator functions like this:

step 1, a liquid refrigerant flows around the refrigerator, evaporates into gas, while absorbing large quantity of heat and hence cool down the interior of the refrigerator; step 2, under pressure the refrigerant gas becomes liquid again and cycles back to step 1.

A superconductor refrigerator works quite similarly actually. Instead of a liquid refrigerant, the step 1 starts with a superconductor, and when the superconductor loses its "superconductivity" and becomes a normal metal, it absorbs a lot of heat and cools the refrigerator; in step 2, the metal is converted back into a superconductor, not through pressure, but through magnetic field, and then the superconductor is ready to cycle back to step 1.

Unfortunately, currently all superconductor requires pretty "extreme" conditions, either at very low temperature or under very high pressure. It is not entirely clear what kind of superconductor is most "convenient" for the purpose of refrigeration, it probably depends on the type of refrigerator.

The superconductor fridge is similar to a conventional refrigerator, in that it moves a material between hot and cold reservoirs. However, instead of a refrigerant that changes from a liquid state to a gas, the electrons in a metal change from the paired superconducting state to an unpaired normal state. Jun 4, 2019

In the superconducting quantum fridge, researchers place a layered stack of metals in an already cold, cryogenic dilution refrigerator:

• The bottom layer of the stack is a sheet of the superconductor niobium, which acts as a hot reservoir, akin to the environment outside a traditional refrigerator
• The middle layer is the superconductor tantalum, which is the working substance, akin to the refrigerant in a traditional refrigerator
• The top layer is copper, which is the cold reservoir, akin to the inside of a traditional fridge.

Conductive refrigeration is simply the transfer of heat from something that you're trying to cool off into a heat sink, an object at lower temperature that can absorb heat, by means of conduction. For example, if you put a hot frying pan into water, then you are cooling it off by having it lose heat into the water.

I don't think I can answer the rest of your questions, as this is such a fundamental property of the second law of thermodynamics: heat will tend to equalize over long time scales. You can never use conduction to cool something off to a colder temperature than that of your heat sink.