Is it permissible for a woman with continuous discharge to perform ablution and go to the mosque? And is she affected by delaying prayer while waiting for the elevator, or obeying her parents, or giving charity? And is she required to check for discharge after returning?

The questioner is warned against obsessive thoughts (waswasah) and yielding to them. Regarding the discharge, if it is continuous or intermittent in a disruptive manner, and there isn't enough time for purification and prayer, then she must perform ablution (wudu') after the time for prayer begins and pray as much as she wishes until that prayer time ends. If the discharge stops for a sufficient time to perform the prayer, then she must wait for the time it stops and pray then. There is no need to investigate its emission, and the default assumption is that it has not exited unless there is certainty. If she doubts its emission, the default is that it has not exited. If she is certain of its emission but doubts its time, then it should be estimated to have occurred after the prayer.