Is it obligatory to dispose of Western products (such as mobile phones and entertainment devices) that support the people of Lot (homosexuals), or is it permissible to use them? And does watching female characters in games fall under the ruling of lowering the gaze?

A company's support for immoral acts does not prevent permissible dealings with it or the purchase of permissible goods from it. Thus, a company's support for homosexuals does not preclude the permissibility of buying a permissible toy and playing with it. As for female characters, if they are images of naked women or incite indecency and arouse desires, then it is not permissible to look at them.