What is church teaching on the souls of animals?
The Church has no such teaching.
I had a Catholic tell me that my dog who had died the day before was a worthless, soulless animal) Isn’t it rather rude to tell someone something like that while they’re grieving?
This Catholic has no basis (in Catholic doctrine) for making this claim (and, yes, it is rude).
However, there is no basis for believing that animals (or plants, or bacteria, or minerals) have any place in the economy of salvation. Only mankind is made in the image and likeness of God, and only humans are endowed with immortal souls.
On this earthly plain it is easy to become attached to house pets (I have cried many tears over various losses throughout the years). But this is not unlike a child becoming attached to a stuffed toy - the child’s love for the toy is no less real or authentic than our love for our pets.
We must not let ourselves fall into the trap of imaging that heaven is a place of earthly fulfillment, where we will be reunited with our beloved pets (or our beloved toys, or our mother’s lasagna, or whatever brought us happiness on earth). These are simply possessions. We must realize that, in heaven, we will be united with Our Lord, and find ourselves in full communion with those who were made in his image and likeness, and that all else will seem - as a child’s toy.
You will not be reunited with your beloved dog in heaven. But you will not feel any loss, and you will remember and appreciate the fleeting happiness your dog brought you while in this life. You will not
need your dog to find happiness in heaven - you will remember and share in this happiness for all eternity (with none of the sorrow).