Is it permissible for my husband to open live broadcasts on TikTok and talk with followers, and to enter into challenges with young men and women, and is the money he earns from the gifts sent to him in these challenges unlawful?

It is not possible to judge the specific actions of the husband. However, in general: Broadcasting permissible or beneficial content is permissible if it is free from Sharia (definition) prohibitions, such as the exposure of 'awrah (intimate parts), the appearance of women who are immodestly adorned (mutabarrijat), or their actions that invite temptation (fitna). If the broadcast is free from such things, then there is no harm for its owner to accept gifts from his followers. As for broadcasting forbidden content that contains Sharia prohibitions, it is not permissible, and it is not lawful to earn money from it or accept gifts because of it. This is because when Allah forbids something, He also forbids its price, and forbidden benefits cannot be exchanged for compensation.