Regarding your cats spirit, I do believe you as many of my departed pets have lingered after death. I cannot say for certain what will happen with your cat. It might be hanging around the house, but it could follow you if its strong enough. My cat was hit outside the house, I moved a year later, and his spirit did not follow. I return to the old house on occasion and I sometimes sense him but not as much now. My dog died on the same property as my cat, but he has followed me through a couple moves. He crossed over about 10 years ago. You might have to consider if your pet is stuck between worlds helping him pass on. I didn't help my dog, he went when he was ready, but sometimes spirits cling to the life they once knew.
In my moves I have invited the spirits who live in the old house to join me in the new house, and try to leave somewhat of an energy trail for them to follow. Again, its their choice. I don't know about an animal spirit, you could try setting up a type of alter for him, with a photo, candle, and some of his favourite treats placed by an open window. Careful not to attract any living animals though.