I noticed that the Imam who leads the Muslims in prayer in Makkah prays toward one direction only, which is that opposite to the door of the honorable Ka‘bah. Why does he not pray towards the other three directions of the Ka‘bah, since, as I know, it is valid to pray towards all four directions?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

The Qiblah for someone who can see the Ka‘bah is the Ka‘bah itself. It is valid for a Muslim to pray towards any direction so long as he is facing the Ka‘bah itself. Moreover, if he prays inside the Ka‘bah, he can face any of its walls, even if he faces the door.

An-Nawawi said, "If he prays inside the Ka‘bah, he can face whatever he wills of its walls. He can also face the door that has a threshold at the height of two thirds a cubit, regardless of whether it is closed or open."

As regards the Imam's standing in the direction opposite to the door of the Ka‘bah nowadays, perhaps it is because the maqam (standing place) of Ibraaheem is located in this direction. Allah Almighty says (what means): {And take [O believers] from the standing place of Ibrahim a place of prayer} [Quran: 2:125]

Qatadah said:

"The believers were commanded to pray beside it, not to pass their hands over it. What is done by the Imams is not an obligation since the Imams in the previous times used to pray in any place of the mosque. Even at the present time, we see the Imam who leads the people in the Taraweeh prayer stand below the sunshade that is located at the direction of Ar-Rukn Al-Yamani." 

Allah knows best.