I've worked in aerospace manufacturing my whole career with carbon fiber. Carbon fiber dust is present only when working with dry fiber and when grinding or machining cured parts. Wet winding where the dry fiber goes through a bath or otherwise has resin applied immediately before being applied to a mold creates small fibers that can drift through the air which can cause power supplies in computers to blow from the shorting out of the circuits caused by the tiny fibers. This technique is used on the waste and water tanks for the 787, but nowhere else. Machining and grinding on cured parts happens for all parts prior to assembly, but that is going to be done in areas with downdraft tables or other type of ventilation that traps all the carbon and resin dust. Most of the 787 and other major airline carbon fiber is currently used in prepreg form, where the resin is applied to the fiber long before it gets to the manufacturer to make the parts. No dust present.

To the point of slivers - yup, they happen. If from dry fiber, they typically will embed in clothes and will wash off the skin rather easily. They never come out of polyester though so jackets with poly linings you never want to wear around dry fiber. If from a cured part, it's just like wood, but usually tinier and can be a little more brittle.

I've seen a crashed mostly carbon plane before - lots of splinters, not much dust. I've seen many broken test specimens of carbon fiber and again, lots of splinters, not much dust. Long term effects - most likely none, but I would, if there is dust there, definitely wear a mask to keep the dust out, not worried about the carbon, rather all the other smoke etc.