Is the prayer valid behind an Imam when the worshipper doubts whether he (the Imam) started the prayer before its prescribed time? And is the 'Isha' prayer valid in this case? What is the ruling if the doubt turns into a strong presumption?

The time for Ishā' prayer begins with the disappearance of the red twilight. It is not a condition for the validity of a calendar that it conforms to the Umm al-Qura calendar. Prayer times are known by their legal signs, and calendars are only used for guidance unless they are based on observation and approved by trustworthy scholars. If there is doubt about the start of the time, the prayer should not be performed unless there is certainty or strong presumption. If the calendar or the Imam are trustworthy, then act according to them. If they are not, then do not pray until you are strongly presumptive that the time has begun.