Is a woman's prayer in an airplane valid when there is no suitable place to pray and no way to determine the Qibla, coupled with the need to use the restroom?

For the prayer to be valid, facing the Qiblah and standing, if one is able, are prerequisites. Among the excuses that permit their omission is the fear of the prayer time expiring if combining the prayers is difficult, or if the prayer is one that cannot be combined [with another].

If the worshipper on the airplane is able to stand and face the Qiblah, then he must do so. If the prayer is one that can be combined, such as Dhuhr with Asr, it is permissible to delay it and combine it upon landing, provided it can be performed before its time expires.

As for purity, it is a condition for the validity of prayer. Praying while there is something distracting, such as gas or urine, is valid but disliked if it does not affect one's devotion (khushu').

If your omission of standing and facing the Qiblah was due to inability, then your prayer is valid. However, if it was done despite being able, then your prayer is invalid and must be repeated.