Is it permissible to mention previous work experience in shops selling alcohol, tobacco, and lottery tickets in one's CV? Is that considered publicizing a sin? And would the income earned based on this experience be unlawful? And is the work considered unlawful if it is through assisting an apostate?

If the activities of businesses vary between what is permissible and what is forbidden, there is no harm in mentioning that one works there, as it is not considered publicizing a sin. However, if its activity is solely forbidden, then its name and forbidden activity must be obscured when mentioning working there, unless there is a need to explicitly state its name. The new job, as long as it is permissible, carries no harm, neither in itself nor in its earnings, because it is in exchange for the current work. As for obtaining a job through mediation, this refers to a previous detailed explanation. And the fact that the person who helped you is an apostate does not affect the permissibility of the job if it is halal.