Does passing gas, which invalidates ablution, prevent one from staying for extended periods in the Haram for Umrah, Hajj, and Itikaf, and is there a way to facilitate that?

If the flatulence is almost constant, then ablution (Wudu') is required at the beginning of the time for each prayer and before circumambulating the Kaaba (Tawaf (definition)), and whatever is released afterward does not matter. If it is not constant, then ablution is required for each prayer or Tawaf, and ablution is invalidated by the release of flatulence. There is no issue with performing I'tikaf (spiritual retreat) with ablution for prayer, Tawaf, and touching the Qur'an. However, reciting the Qur'an or touching it with a barrier does not require ablution.