Is it permissible to complete one's studies at an English language institute whose curriculum includes music and indecent pictures of foreign women, knowing that the institute does not refund the money paid, and can the institute be sued for not adhering to the agreed-upon attendance conditions?

We hope that there is no problem for you in joining this institute and continuing in it, especially since it is possible to avoid the two forbidden things: pictures and music. As for pictures, you can refer to Fatwa No. 142589. As for music, it is said concerning it: "So fear Allah as much as you are able" [At-Taghabun: 16], keeping in mind that what is forbidden is the intention to listen, not merely hearing. You can refer to Fatwa No. 146713.