Islam teaches its followers to be kind to all creatures except those that are harmful by nature such as snakes, scorpians etc...
However, it always distinuishes humans from all other creatures because human have been given this preference by Allah over animals. This distinction is shown in many places. One of those times is the time of death. When a human being dies, the process of funeral and burial is completely different than that of others. When an animal dies, Islam teaches us to simply dispose off the corpse properly and it does not extend the same rites to the animal regardless of how dear the animal may had been to its owner or caretaker.
Also, Islam does not allow cremation of any living creatures. So cremating an animal or a human being after death are both forbidden.
Now, I will answer your questions orderly Insha'allah:
1- Washing, shrouding and proper burying of a dead animal is not in accordance with the teachings os Islam and cremating is not allowed either under ordinary circumstances.
2- Islam does not negate that there would be pets/animals in Paradise.
3- Animals do not have any accountability. Therefore, they do not go though any thing similar to the humans go though after their death.
And Allah knows best.
Mufti Ikram ul Haq
Fatwa Center of America
8 Ramadhan 1432/ August 8 2011