What is the ruling on performing ablution (wudu') and ritual bathing (ghusl) with Zamzam water?

It is permissible to perform ablution and ritual bathing with Zamzam water. However, scholars disliked using it for cleaning after relieving oneself (istinja'). And Allah the Almighty knows best.

A person prays in a certain spot within the rows of the mosque, and gets angry if someone else prayed at his spot. What is the ruling on this behavior?

All Perfect Praise be to Allah The Lord of The Worlds and may His Peace and blessings be upon our Prophet Muhammad and upon all of his family and companions                                                                                                                                                                                                                              Prophet Mohammad (PBUH) make it prohibited for a Muslim from having/taking a certain spot in the mosque when performing prayers if they arrived and find it occupied, furthermore; it is the religious duty on others (praying people) to offer a piece of advice to alert the ones who don't know, that whoever reaches a spot first at the mosque having a priority/privilege upon others in that spot. And Allah Knows Best.

Is it permissible to combine and shorten prayers at home before traveling?

It is impermissible to combine and shorten prayers at home before traveling because a person is called a traveler once he/she departs the borders of his/her country, and offering prayer that way was originally intended to exempt a traveler. And Allah Knows Best.

Which is more excellent for a woman: to pray in the mosque or in her house?