Should we assume that animals have spiritual feelings?
I would be surprised if they didn’t. After all, Christians believe that the Spirit indwells creation, so it is reasonable to suppose that sentient animals have a relationship with God. If that sounds strange, it is only because we are still in thrall to the Cartesian notion that animals are machines with no interior life.
Should you grieve for an animal differently than for a person?
Grieving is grieving; bereavement is bereavement. The process is the same for companion animals as it is for humans. Each stage of bereavement (shock, denial, despair, etc) is present in both cases.