Double processing is where one takes the RAW and processes it twice. First time is done normally, whereby you set the values to what you normally would choose for the photo as a whole. Then the second time you process the RAW you keep all the values the same, but you lower the exposure slider so that the overexposed areas are pulled back out of the limit (and if they only clipped hopefully also restores the lost detail - something that the burn tool can't do).
You then just take each version of the RAWs in the editing software and paste the latter over the former, using a layermask so that only the areas that you want are shown (note you might also want to lower the opacity of the brush whend doing this, around 10 is good as you can then build up the revelation of the key areas in slow steps, letting you choose just how much you want to show).
Not if you are using photoshop (or elements) you will have to save the first version of the processed RAW under a different name as it originally was called before you open the RAW again, otherwise it won't work correctly (as it will detect that you've already got that photo open and thus won't be allowed to work on it).
Artifacts won't be caused by dust on the sensor, but they are a common byproduct of oversharpening a photo. I'd pull back on the sharpening to reduce how many artifacts it generates; if you want more sharpening you'll have to work harder with layermasks and selective application with sharpening. Dust on the sensor typicallly creates what are known as dust spots, these are blurry black/dark spots that will appear dotted over the photo.