Is it permissible for a woman to insert an IUD herself without medical supervision, and would the female doctor be sinful and her earnings unlawful for inserting IUDs for women, knowing that the IUD is forbidden due to the exposure of the 'awrah?

It is permissible for a woman to have an IUD inserted by a female doctor if there is a need, and no other means of contraception are available. This is because the insertion of an IUD is among the needs that permit the exposure of a woman's awrah (private parts), and because a woman looking at another woman is less severe than a man looking at her. The female doctor is allowed to charge a fee for this.