Nezuko is initially 12 years old in the manga. As the story unfolds, her age doesn't change significantly in a traditional sense, but her character development is quite remarkable.
Hot water burns would cause a burning pain. After the hot water came into contact with the skin, the heat would be quickly transferred to the skin cells, causing thermal damage. Although burns might only affect the superficial layer of the skin, the injured would feel a burning pain because of the rich nerve endings in the skin, which made them sensitive to pain. When dealing with burns, you can rinse or soak the wound with running cold water, or you can use ice to apply cold compress to the area to reduce pain and swelling. It was not recommended to directly touch the ice on the skin to avoid low temperature damage. Under the doctor's guidance, you can apply an appropriate ointment to promote wound recovery. There may be short-term skin hyperchromation after burns, but it usually does not affect normal life and will gradually subside.
A simple Nezuko and Zenitsu fanfiction could be that they go on a journey together. Along the way, they encounter other demon slayers who are initially skeptical of Nezuko. But Zenitsu stands up for her, explaining that she's different. This shows the depth of their friendship and how Zenitsu has come to trust Nezuko completely.