Is designing websites—that do not contain images of women or indecent images, and do not include music or songs—peralmissible (halal), and is the money earned from it permissible? And is the designer sinful if the client later places prohibited things on the site? And is it permissible to include images of children in the design?

Given the prevalence of sins and the diversity of their methods, it is not permissible to cooperate in sin and aggression. If the seller, designer, or producer is certain that what they are selling or designing will be used for unlawful purposes, then it is not permissible for them to sell or produce it. This includes selling grapes to someone who will use them to make wine, selling copper to someone who will use it to make a bell, and anything else where it is known that the buyer intends to use it for a forbidden purpose.

However, if the seller or designer doubts or does not know anything about the eventual use of what they will sell and produce, then there is no harm upon them in selling and designing it, and the sin falls upon whoever uses it for unlawful purposes.

Regarding website design, if you know that the client wants to design a website to use it for unlawful purposes (such as usurious banks, pornographic websites, selling forbidden items, or movie and music websites), then it is not permissible for you to design the website for them. However, if you do not know anything about the reason for the request to design the website, or if the website is predominantly used for permissible purposes, then there is no harm upon you in designing and selling it.

As for the ruling on including children's images on the websites you design, this is not permissible, because photography is forbidden except for what is necessitated by necessity or need.