Nezuko remains alive in the Demon Slayer manga. The author chose to keep her alive as she is an important character with a vital part to play in the overall narrative.
Well, Nezuko passes Sanemi's test through a combination of factors. Firstly, her strong willpower helps her stay calm and composed. Secondly, her bond with Tanjiro gives her the strength to overcome the challenges. Finally, her inherent kindness and desire to protect others shine through, convincing Sanemi.
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.