Is it permissible for a woman to wear loose-fitting clothes that are not transparent and cover the body, without trousers underneath, with the possibility of her feet showing in some cases such as jumping or climbing stairs, or must she wear loose-fitting trousers underneath to ensure complete covering?

If a woman's clothing covers her entire body, is not form-fitting, not transparent, and not adorned in itself, then it is a legitimate hijab (definition), regardless of its name or appearance. The legitimate hijab does not require a woman to wear trousers underneath it. Rather, anything that achieves concealment for the woman—including her feet—and fulfills the Sharia (definition) requirements, is permissible for her to wear. If a woman needs to ride a vehicle or ascend (stairs, etc.), or similar situations that might expose her feet or shins, then she must take precautions and wear trousers. Likewise, one should be wary of obsessive doubt (waswasa) in this matter.