Sorry I missed this. If the probe has some stuck bits of meat on it, wrap the probe in a damp paper towel. Just the part that has the stuck bits and I mean damp, not wet. As for washing, I damp paper towel with a drop of dish soap and then rinsed with a wetter towel and dried off. Very important to not get the area where the wires enter wet. And do not soak your probe.
As for the staining, just smoke stain and its components. Sort of like creosote and ugly but does no harm. I just leave mine alone.