Really depends on what you are mining for, and what you are mining and, how you are mining.
In 0g environment even weak and heavy materials can be used for functional constructions. As an example collecting the energy, meted "rock" coated with metal can be used as mirrors for solar collectors. So there might be no waste at all in the sense. Gravity dictates for construction materials to be strong enough, not the case in 0g.
The technology you are using also important for determining in which form those byproducts might be but as rule of thumb quite the same as here on earth for the materials you aiming for.
Most likely you need to feed your processing facility with small chunks of the asteroid, and depending on the technology of extraction it might be gravel or dust-like particles, but it does not mean that output waste will be those particles or small chunks.
As an example, different floating methods are used to separate and concentrate metal ores, and input is powder like stuff lot of water and lot of bubbles, but if there not so much water on that particular asteroid you have to squeeze as much as possible from the waste, to not loose your water. It means some postprocessing sequence for the waste, which might or might not include forming bricks from the waste at the end.
I will not wonder if it will be easier to smelt and skim the stuff you need and leave the big melted rock, for later use.
No, it's not become a navigational hazard to the equipment. Space is big and all asteroid combined aren't that big compared to space, and on close orbits, the speed difference isn't that high, and to change the situation you have to throw the stuff really hard, which is in fact waste of energy.
But in general, the mining construction have to incorporate some solutions against meteorites, as it will face the problem on the way and on the place, just because there are meteorites flying around.
Again depends on the body where you are mining, and how you are doing that.
On the Moon, there way much more dust than you will be able to produce and it definitely needs a proper design for the equipment. Same seems to be true for and large size asteroid. But there are different ways to do a proper design and it has not to be a big deal.
The dust is considered to be a good source material because it is already in a form suitable for different methods of separation of the raw ores, so it might be so that the dust will be the target of the mining equipment or something like that. The point is that it most likely will not fly freely around.
If you consider using solar panels as the main source of energy, then there is another important factor to consider - energy needed to process materials is pretty high, so the area where you dig will be small (at any given moment) compared to the solar panel surface and face in opposite direction to the source of the dust.
But again heavily depends on the way you dig and process the materials, a thing which has to be taken into consideration but not a big source of problems by itself.
clogging servos - Produce a Hermetic Seal with Ferrofluids might solve all your problems in that regard.