Is the Ka'ba situated in the center of the world?

Geographically, yes, the Ka'ba is located at the center of the world. If we have a sphere, then any point upon that sphere is the center.

When I accepted Islam, some friends of mine said that the birds will not even fly over the Ka'ba, instead they will fly around the Ka'ba. The forth field prevents them flying over the Ka'ba.

However, when I went to perform Umrah, I saw birds flying over the Ka'ba. When it rains in Makkah, people bring bottles to get water from the Ka'ba. They will get their robes drenched which they are planning to get buried with, and they will wash it with Zam-Zam water. However, the fact is that the water is filled with filth (najasa). This is distortion, misunderstanding and deviating from the practical and real aspects.

Hence, not only Ka'ba, but Masjid Nabawi, Majid Al-Aqsa and even our home is the center of the sphere.