First please accept my condolences for the death of your loved one. I believe that animals are spirit beings that deserve the same respect and love that all spirit beings are worthy of.
The ancient Greek world had established belief in an afterworld and the ceremonies surrounding death by the 6Th century BCE. Homer wrote of an underworld where Hades, the brother of Zeus and Poseidon, and his wife, Persephone, reigned over lost and departed souls.
We believed at the moment of death the spirit or psyche left the body as a little puff of wind or breath. We concentrated on honoring the dead with burial rites that honored and exalted the dignity and spirit of the deceased. We mourned our departed loved ones with songs, poems, sculptures and monuments. Once buried, we continued to remember and honor our loved ones with small cakes and libations when visiting their graves.
Burial rites were so important to the soul’s afterlife that refusing them was considered a type of punishment. When the free soul leaves the body it begins its afterlife as a spirit. We believed that the spirit retained its physical appearance and psychological attributes.
Although we did not believe in reincarnation per se we did believe in hero worship. Heroes were given special attributes and powers that lived on after death. This becomes personal as each of us has different definitions and perspectives on what constitutes an hero. Personally, I believe in the evolution and reincarnation of the spirit through out eternity and hope to see my departed loved ones again in some form. I won’t say the future - because for all I know - time may not be linear and reality may be an illusion.
I send my sincere sympathies for your loss. I think the underworld is a time of refection and preparation for the higher worlds. Departed spirits leave a trail of love that we continue to follow and revere on our pathway through the stars.