What is the validity of the fast of one who missed a prayer while trying to relieve herself with excessive speed, to the extent that water entered her stomach? And does she have to make up the missed prayer?

We ask Allah to grant you well-being and guidance. It appears that the waswas (whispers/obsessive thoughts) have reached a significant level, causing you to delay prayers beyond their prescribed times. The reason for this is your yielding to these waswas. The cure for waswas is to turn away from them and pay them no mind. Your fast is valid as long as the water entered your stomach unintentionally. You must make up the missed prayer if you have not already prayed it.