To answer the question of what happens when a god dies, we first need to define all components of a god. Gods have three components.
The energy of that god. This could be a feeling or traces they leave behind. The deity, which appears in a form that interacts with the living. The things they represent. This can be love, gambling, or anything within their sphere. Sometimes, they are philosophies or parts of the brain. Some are thoughtforms or egregores.
Keeping in mind that the gods we see and interact with are just part of a cosmic presence. Let’s say that you attempted to murder a god of nature. This would be only killing the messenger form of that deity because they are nature. You would need to get rid of nature in order to remove their existence from the world. Gods of these huge concepts are basically unkillable in practice, because even if they do not visit us, the things they represent cannot be removed from existence. New gods and other deities not connected to anything within our world are killable, but even this is difficult. Very few new gods have ever died on the spiritual level, even those that are the product of a god and a human are invulnerable. When a god dies on a spiritual level, it is the equivalent of a human dying, because energy cannot be destroyed so it just gets recycled to become better. Evil gods that have been slain such as The Sarin God would have been sent temporarily to the higher planes to improve their state of existence, or even the punishment pits.
Despite this though, it is a lot more common for evil spirits and gods to be sealed away in tombs or banished from certain planes of existence. Actually killing deities is hard even for other deities, because they are invulnerable. It is more common for them to choose to retire from godhood and ascend into nirvana. When they do this, they are always able to return into existence when they are ready. However, some ancient deities such as Biachar or The Lady Of The Waves, formerly Darmala have had too much of an effect on the world to ever retire. Even if the ocean or the moon were destroyed entirely, the world would still have water and lunar energy, so these deities will exist until the end of time or if they choose to retire for whatever reason.
When a god dies, another consciousness has to be nearby to ascend in its place. This does not need to be the person who killed the god or even have any connection to it. A human can theoretically become a god by standing witness to the death of a god and rising in its place to replace them, but this is even more rare than the death of a god. In cases like these, characteristics from the new ascended person combine with the traces of the dead god to create a new deity. The God of Fear And Hunger is a good example of this, since she ascended by eating the flesh of a half-god.
Praying to a god that has died can revive them in some cases, but regardless their traces continue to affect reality. Some deities may be too ancient to be currently active, but they can be re-awakened with energy and offerings as well as worship. Archaic deities may need the blood sacrifice of an animal to be reawakened, but the success of this cannot be guaranteed as it is hard to get sigils or channeled messages from sleeping deities. Awakening sleeping gods also directs all of their attention directly towards you, which can be seen as either a positive or a negative. It is best to form a following of people for the deity before awakening them, so they do not need to return to slumber.