Is the "greeting of the mosque" prayer (Salat Tahiyat Al-Masjid) obligatory in the courtyards of the Grand Mosque in Mecca and the Prophet's Mosque in Medina when one is unable to pray inside the mosque itself? And does one's prayer in the Haram become void if someone passes in front of them due to the crowd?

Tahiyyat al-masjid (greeting the mosque prayer) is a confirmed Sunnah according to the majority of scholars. For one who is unable to enter the Grand Mosque or the Prophet's Mosque due to overcrowding, he should pray it in the courtyard belonging to it. The passing of one of the worshipers in front of you does not invalidate the prayer, whether you are in the Haram or elsewhere.