Are black ash particles and holes on a cleaning sponge normal?
Basically what you have there is a standard household cleaning sponge. These are made from either Polyurethane or Polyester (most likely the latter). Neither of these materials hold water very well because of their structure and material. This means the material will not be cooled by the water and so will be exposed to the high temperatures of the soldering iron tip. They are not designed to withstand the heat of soldering - for example polyester sponges will melt at 250*C.

Sponges used for soldering are made from compressed cellulose. This is much denser and critically will hold water and stay damp (you want the sponge damp, not wet!). If it is kept damp the sponge will withstand the temperature of soldering because the water inside it cannot be heated above boiling point - it turns to steam if you try. Now if you were to hold the soldering iron on it for an extended period of time, the water would eventually evaporate away and the sponge would melt, but you don't normally hold it there long enough to do that.

Googling "Compressed Cellulose Sponge" or "Soldering Sponge" will find you many results of where to buy the correct sponges.