"I am faced with a very sick cat I have had for 19 years. He is a part of my family and I am feeling greatly sad. I know my pet is not human and probably not considered Jewish, but are there any "Jewish" issues with 1) ending his suffering, and 2) cremation or burial? I hope you do not think these questions are disrespectful."
However, as with family loss, we have to realize that it is part of life. We lose relatives and devoted pets, but they move out of a suffering body into a state of eternal rest.
It might be wise to purchase another cat, similar in breed to the previous one, and even name it the same name as the previous one. This will somewhat alleviate the pain.
This all teaches us a lesson. We should strive to do our duty in this world. Be altruistic, devoted and caring to friends, neighbors animals, etc. No creation should be treated with disdain. We should envision this world a vestibule and entrance to the afterworld. Kindness and good deeds will merit our entrance into this world of eternity - (source: "Ethics of the Fathers" 4:16).
As to "ending his suffering" - one may and should do so as soon as the animal has no chance of recovery and is only suffering - (source: "Code of Jewish Law" E.H. 5:14).
Once an animal is dead, burial or cremation is permitted - (source: Exodus 22:30).
With blessings from Jerusalem,
Rabbi Shraga Simmons