Mining on earth usually consists of 2 or 3 stages.
First, the ore is ground up - hence the previously referenced steel balls. Put soft-ish rocks in a steel drum with a bunch of steel balls and gravity will help pulverize them to a powder. (See any problem yet?) There are rollers and jaw crushers that can break up the bigger bits (is how gravel is made) but nothing says “make mine powder” like heavy steel balls or rods in gravity. Of course, you add water to help with the grinding, and help carry away any heat generated. (See any other problems?)
For gold, this fine powder can then be given the industrial equivalent of “panning”, where stuff is sloshed around on a grooved vibrating table so gravity separates the heavier gold particles from the lighter rock. (Hmm, gravity again… and water)
Next, the separated concentrated gold flakes are dried (something actually much easier in space, if you don’t mind throwing away water) and melted so that gravity (again) separates the slag from gold. Slag of molten rock floats over the molten gold which is much denser.
For other metals, the principles are not much different - except we may add an electroplating step to refine the metal even further.
Given the right equipment, a spinning environment will substitute for gravity - but this requires a much larger piece of equipment. Something like the spinning wheel from 2001, but able to handle hundreds of tons of material pounding away all the time. It would be a major construction. Moving stuff in space with rockets gets very expensive, so I imagine it would be moored to the asteroid being eaten, so material could be moved with electric motors and conveyor belts or such. It would take decades or more to eat through something a few miles in diameter. you only want to send the finished product home where it needs rocket fuel.
Melting stuff is easy - solar reflectors - except how do you contain the result? Perhaps you could have solar cells and run electricity through the ore to melt it - we’re talking, again, some serious size solar cell farm.
In The Moon Is a Harsh Mistress Heinlein describes a linear motor acceleration launcher, an efficient way to “fire” a load toward earth where presumably it would be captured by a local tug craft and guided to where it was needed.
One concept would be to cast ingots in the shape of re-entry craft, and if some of the material burns off during re-entry, so what? Until the scale of metal production is so high we all start worrying about fine particulate metal oxides and nitrides in the atmosphere created by ablation of all those re-entry craft. maybe we could land them in Antarctica or Greenland where they cool off easily.
Basically, at this point we’re still decades and orders of magnitude away from industrial mining in space.