According to certain rules of physics, a toroid planet can exist, however the existence of a planet in nature is very unlikely.
For gravity to work, the angular momentum and mass must be large enough. The planet will not become a disc in space because of the centrifugal forces spreading out the forces. Think of clothes in the washing machine spreading out as the rotations speed up. The surface gravity depends on location. It is weakest along the interior and exterior equator, while strongest slightly hubward from the “poles.” This can be a fairly major difference.
Since light is determined by the degree of tilt and size of the planet, the atmosphere is greatly affected by these too. The Coriolis Effect makes air moving towards or away from the rotational axis bend away, since it has more or less velocity than the ground. A parcel of air “at rest” near the equator has a lot of actual momentum since the equator is moving fast around the rotation axis: if that air were to flow pole-wards it would now have a noticeable velocity eastwards or westwards. This is why the global airflow is not just simple convection cells from the equator towards the poles: as heat is transferred using air polewards the air flow gets twisted around, producing trade winds.
A moon is even possible with a doughnut planet. However the orbit will vary every day but may, in fact, become a cycle. It is even possible for the moon to orbit through the hole of the planet if it is large enough.