Do any satellites have an orbital period greater than 24 hours? Can an orbit have an apogee greater than 35768 kms?

Yes, there are satellites with a period greater than 24 hours and yes, there are orbits with their apogee larger than 35,768 km.

Orbits above geosynchronous orbit are referred to as High Earth Orbit (HEO). Examples of satellites in those orbits are the Vela satellites. There were twelve Vela satellites. They were designed to detect nuclear explosions on or above Earth, after the Partial Test Ban Treaty was passed in 1963. The Vela satellites are no longer functioning, but they are still orbiting Earth. For example, Vela 12 has a perigee of 37,174 km and an apogee of 185,641 km.

A fairly recent satellite launched into HEO is the IBEX (Interstellar Boundary Observer). In order to do its job, it needs to be higher than the Earth’s magnetosphere. It has a perigee of 81,742 km and an apogee of 277,747 km.

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