Yes, get a breathable car cover
you won't suffer the problems of the cover scratching anything because the garage will stop the wind from lifting it.
You could also hang a tarp / sheet in the garage to catch any fall out. You could do this if your worried about covering the car.
Is the roof all wood, single board (large section boards) or slats and is there any joists etc etc. Thinking metal contaminant but may well be sap or bitumen, combined with the dust particles.
Thing that makes it worse in a garage is the heat, anything that sits on the car will bake on.
Would go for a cover myself, doesn't need to be a fitted one, a loose fit one will be fine for inside.
Sounds like you need to soften first, snow foam, Citrus and or TFR, maybe G101 or Tardis (dabbed on) clean the car and Clay bar. If they are stubbonly stuck, you may need to go for a de-nibber and then machine polish to finish down.
Worst case senario would be to wet sand but I don't think it would be that bad.
Work small areas with short motion to minimise scratching, should be able to polish out any marring or fine scratches if necessary.